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Metamath Proof Explorer

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 ) )