Theorem canth3

Description: Cantor's theorem in terms of cardinals. This theorem tells us that no matter how large a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of Monk1 p. 133. (Contributed by NM, 5-Nov-2003)

	Ref	Expression
Assertion	canth3	⊢ ( 𝐴 ∈ 𝑉 → ( card ‘ 𝐴 ) ∈ ( card ‘ 𝒫 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		canth2g	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴 )
2		pwexg	⊢ ( 𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V )
3		cardsdom	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ V ) → ( ( card ‘ 𝐴 ) ∈ ( card ‘ 𝒫 𝐴 ) ↔ 𝐴 ≺ 𝒫 𝐴 ) )
4	2 3	mpdan	⊢ ( 𝐴 ∈ 𝑉 → ( ( card ‘ 𝐴 ) ∈ ( card ‘ 𝒫 𝐴 ) ↔ 𝐴 ≺ 𝒫 𝐴 ) )
5	1 4	mpbird	⊢ ( 𝐴 ∈ 𝑉 → ( card ‘ 𝐴 ) ∈ ( card ‘ 𝒫 𝐴 ) )