DefPowerfulClass (3.45, Definition): “Is powerful” predicate

Standalone kernel entity record from 3. Set/Class-Theoretic Foundations (3.01–3.78) — View in Complete Entity Database (#DefPowerfulClass) →

Formula Display:
Proof Trees:

DefPowerfulClass (3.45) — “Is powerful” predicate

Morse-Kelley / Von Neumann-Bernays-Gödel class comprehension, sethood, ordinals, Dunaev pairs, cardinals, universes, and reflection.

Num Entity Database Record (Vertical Field Table)
3.45
Title “Is powerful” predicate #DefPowerfulClass #label-is-powerful-predicate
Classification Definition
Formula
0(∀(U:Class)(Powerful(U)⟷(∀(B:Class)((B∈U)→(𝒫(B)∈U)))))
0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/forall-binder.html}{\forall} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{U} \href{../symbols/typing-colon.html}{:} \href{../symbols/class-type.html}{\mathsf{Class}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/powerful-pred.html}{\mathsf{Powerful}} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{U} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/iff-conn.html}{\longleftrightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/forall-binder.html}{\forall} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{B} \href{../symbols/typing-colon.html}{:} \href{../symbols/class-type.html}{\mathsf{Class}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{B} \href{../symbols/in-pred.html}{\in} \href{../symbols/variables.html}{U} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/power-op.html}{\mathcal{P}} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{B} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/in-pred.html}{\in} \href{../symbols/variables.html}{U} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST PKernelDef_prf
Scheme of Types 0(∀([U@Variable]:[Class@Type])(Powerful([U@Class.Expression])⟷(∀([B@Variable]:[Class@Type])(([B@Class.Expression]∈[U@Class.Expression])→(𝒫([B@Class.Expression])∈[U@Class.Expression])))))
Verification Definitional (Kernel AST) · Standalone Entry (DefPowerfulClass) → · (3.45) · #DefPowerfulClass