Theorem domen1

Description: Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003)

	Ref	Expression
Assertion	domen1	$$⊢A\approx B\to \left(A\preccurlyeq C\leftrightarrow B\preccurlyeq C\right)$$

Proof

Step	Hyp	Ref	Expression
1		ensym	$$⊢A\approx B\to B\approx A$$
2		endomtr	$$⊢\left(B\approx A\wedge A\preccurlyeq C\right)\to B\preccurlyeq C$$
3	1 2	sylan	$$⊢\left(A\approx B\wedge A\preccurlyeq C\right)\to B\preccurlyeq C$$
4		endomtr	$$⊢\left(A\approx B\wedge B\preccurlyeq C\right)\to A\preccurlyeq C$$
5	3 4	impbida	$$⊢A\approx B\to \left(A\preccurlyeq C\leftrightarrow B\preccurlyeq C\right)$$