Metamath Proof Explorer


Theorem cardidg

Description: Any set is equinumerous to its cardinal number. Closed theorem form of cardid . (Contributed by David Moews, 1-May-2017)

Ref Expression
Assertion cardidg ( 𝐴𝐵 → ( card ‘ 𝐴 ) ≈ 𝐴 )

Proof

Step Hyp Ref Expression
1 elex ( 𝐴𝐵𝐴 ∈ V )
2 cardeqv dom card = V
3 2 eleq2i ( 𝐴 ∈ dom card ↔ 𝐴 ∈ V )
4 cardid2 ( 𝐴 ∈ dom card → ( card ‘ 𝐴 ) ≈ 𝐴 )
5 3 4 sylbir ( 𝐴 ∈ V → ( card ‘ 𝐴 ) ≈ 𝐴 )
6 1 5 syl ( 𝐴𝐵 → ( card ‘ 𝐴 ) ≈ 𝐴 )