Theorem domentr

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

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

Proof

Step	Hyp	Ref	Expression
1		endom	$$⊢B\approx C\to B\preccurlyeq C$$
2		domtr	$$⊢\left(A\preccurlyeq B\wedge B\preccurlyeq C\right)\to A\preccurlyeq C$$
3	1 2	sylan2	$$⊢\left(A\preccurlyeq B\wedge B\approx C\right)\to A\preccurlyeq C$$