Theorem isnumi

Description: A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015)

	Ref	Expression
Assertion	isnumi	⊢ ( ( 𝐴 ∈ On ∧ 𝐴 ≈ 𝐵 ) → 𝐵 ∈ dom card )

Proof

Step	Hyp	Ref	Expression
1		breq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵 ) )
2	1	rspcev	⊢ ( ( 𝐴 ∈ On ∧ 𝐴 ≈ 𝐵 ) → ∃ 𝑥 ∈ On 𝑥 ≈ 𝐵 )
3		isnum2	⊢ ( 𝐵 ∈ dom card ↔ ∃ 𝑥 ∈ On 𝑥 ≈ 𝐵 )
4	2 3	sylibr	⊢ ( ( 𝐴 ∈ On ∧ 𝐴 ≈ 𝐵 ) → 𝐵 ∈ dom card )