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	⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐵 ≺ 𝐶 ) → 𝐴 ≺ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		endom	⊢ ( 𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵 )
2		domsdomtr	⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶 ) → 𝐴 ≺ 𝐶 )
3	1 2	sylan	⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐵 ≺ 𝐶 ) → 𝐴 ≺ 𝐶 )