Theorem endomtr

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

	Ref	Expression
Assertion	endomtr	⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝐶 ) → 𝐴 ≼ 𝐶 )

Proof

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