Theorem sbthb

Description: Schroeder-Bernstein Theorem and its converse. (Contributed by NM, 8-Jun-1998)

	Ref	Expression
Assertion	sbthb	⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴 ) ↔ 𝐴 ≈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		sbth	⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴 ) → 𝐴 ≈ 𝐵 )
2		endom	⊢ ( 𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵 )
3		ensym	⊢ ( 𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴 )
4		endom	⊢ ( 𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴 )
5	3 4	syl	⊢ ( 𝐴 ≈ 𝐵 → 𝐵 ≼ 𝐴 )
6	2 5	jca	⊢ ( 𝐴 ≈ 𝐵 → ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴 ) )
7	1 6	impbii	⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴 ) ↔ 𝐴 ≈ 𝐵 )