Metamath Proof Explorer


Theorem ficardid

Description: A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013)

Ref Expression
Assertion ficardid ( 𝐴 ∈ Fin → ( card ‘ 𝐴 ) ≈ 𝐴 )

Proof

Step Hyp Ref Expression
1 finnum ( 𝐴 ∈ Fin → 𝐴 ∈ dom card )
2 cardid2 ( 𝐴 ∈ dom card → ( card ‘ 𝐴 ) ≈ 𝐴 )
3 1 2 syl ( 𝐴 ∈ Fin → ( card ‘ 𝐴 ) ≈ 𝐴 )