Theorem cardid

Description: Any set is equinumerous to its cardinal number. Proposition 10.5 of TakeutiZaring p. 85. (Contributed by NM, 22-Oct-2003) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Hypothesis	cardval.1	⊢ 𝐴 ∈ V
Assertion	cardid	⊢ ( card ‘ 𝐴 ) ≈ 𝐴

Proof

Step	Hyp	Ref	Expression
1		cardval.1	⊢ 𝐴 ∈ V
2		numth3	⊢ ( 𝐴 ∈ V → 𝐴 ∈ dom card )
3		cardid2	⊢ ( 𝐴 ∈ dom card → ( card ‘ 𝐴 ) ≈ 𝐴 )
4	1 2 3	mp2b	⊢ ( card ‘ 𝐴 ) ≈ 𝐴