Theorem sbthb

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

	Ref	Expression
Assertion	sbthb	$$⊢\left(A\preccurlyeq B\wedge B\preccurlyeq A\right)\leftrightarrow A\approx B$$

Proof

Step	Hyp	Ref	Expression
1		sbth	$$⊢\left(A\preccurlyeq B\wedge B\preccurlyeq A\right)\to A\approx B$$
2		endom	$$⊢A\approx B\to A\preccurlyeq B$$
3		ensym	$$⊢A\approx B\to B\approx A$$
4		endom	$$⊢B\approx A\to B\preccurlyeq A$$
5	3 4	syl	$$⊢A\approx B\to B\preccurlyeq A$$
6	2 5	jca	$$⊢A\approx B\to \left(A\preccurlyeq B\wedge B\preccurlyeq A\right)$$
7	1 6	impbii	$$⊢\left(A\preccurlyeq B\wedge B\preccurlyeq A\right)\leftrightarrow A\approx B$$