Theorem cardennn

Description: If A is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013)

	Ref	Expression
Assertion	cardennn	⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ ω ) → ( card ‘ 𝐴 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		carden2b	⊢ ( 𝐴 ≈ 𝐵 → ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) )
2		cardnn	⊢ ( 𝐵 ∈ ω → ( card ‘ 𝐵 ) = 𝐵 )
3	1 2	sylan9eq	⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ ω ) → ( card ‘ 𝐴 ) = 𝐵 )