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Metamath Proof Explorer

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 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		pweq	⊢ ( 𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴 )
2		breq12	⊢ ( ( 𝑥 = 𝐴 ∧ 𝒫 𝑥 = 𝒫 𝐴 ) → ( 𝑥 ≺ 𝒫 𝑥 ↔ 𝐴 ≺ 𝒫 𝐴 ) )
3	1 2	mpdan	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ≺ 𝒫 𝑥 ↔ 𝐴 ≺ 𝒫 𝐴 ) )
4		vex	⊢ 𝑥 ∈ V
5	4	canth2	⊢ 𝑥 ≺ 𝒫 𝑥
6	3 5	vtoclg	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴 )