Theorem carden2

Description: Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden , the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013)

	Ref	Expression
Assertion	carden2	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) ↔ 𝐴 ≈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		carddom2	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( ( card ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ↔ 𝐴 ≼ 𝐵 ) )
2		carddom2	⊢ ( ( 𝐵 ∈ dom card ∧ 𝐴 ∈ dom card ) → ( ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐴 ) ↔ 𝐵 ≼ 𝐴 ) )
3	2	ancoms	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐴 ) ↔ 𝐵 ≼ 𝐴 ) )
4	1 3	anbi12d	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( ( ( card ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ∧ ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐴 ) ) ↔ ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴 ) ) )
5		eqss	⊢ ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) ↔ ( ( card ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ∧ ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐴 ) ) )
6	5	bicomi	⊢ ( ( ( card ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ∧ ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐴 ) ) ↔ ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) )
7		sbthb	⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴 ) ↔ 𝐴 ≈ 𝐵 )
8	4 6 7	3bitr3g	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) ↔ 𝐴 ≈ 𝐵 ) )