Theorem ensdomtr

Description: Transitivity of equinumerosity and strict dominance. (Contributed by NM, 26-Oct-2003) (Revised by Mario Carneiro, 26-Apr-2015)

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

Proof

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