Theorem domentr

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

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

Proof

Step	Hyp	Ref	Expression
1		endom	⊢ ( 𝐵 ≈ 𝐶 → 𝐵 ≼ 𝐶 )
2		domtr	⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶 ) → 𝐴 ≼ 𝐶 )
3	1 2	sylan2	⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶 ) → 𝐴 ≼ 𝐶 )