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	⊢ ( 𝐴 ∈ 𝐵 → ( card ‘ 𝐴 ) ≈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ 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	⊢ ( 𝐴 ∈ 𝐵 → ( card ‘ 𝐴 ) ≈ 𝐴 )