Theorem canth2g

Description: Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of Suppes p. 97. (Contributed by NM, 7-Nov-2003)

	Ref	Expression
Assertion	canth2g	$$⊢A\in V\to A\prec 𝒫A$$

Proof

Step	Hyp	Ref	Expression
1		pweq	$$⊢x=A\to 𝒫x=𝒫A$$
2		breq12	$$⊢\left(x=A\wedge 𝒫x=𝒫A\right)\to \left(x\prec 𝒫x\leftrightarrow A\prec 𝒫A\right)$$
3	1 2	mpdan	$$⊢x=A\to \left(x\prec 𝒫x\leftrightarrow A\prec 𝒫A\right)$$
4		vex	$$⊢x\in \mathrm{V}$$
5	4	canth2	$$⊢x\prec 𝒫x$$
6	3 5	vtoclg	$$⊢A\in V\to A\prec 𝒫A$$