Theorem endomtr

Description: Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998)

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

Proof

Step	Hyp	Ref	Expression
1		endom	$$⊢A\approx B\to A\preccurlyeq B$$
2		domtr	$$⊢\left(A\preccurlyeq B\wedge B\preccurlyeq C\right)\to A\preccurlyeq C$$
3	1 2	sylan	$$⊢\left(A\approx B\wedge B\preccurlyeq C\right)\to A\preccurlyeq C$$