Theorem sdomentr

Description: Transitivity of strict dominance and equinumerosity. Exercise 11 of Suppes p. 98. (Contributed by NM, 26-Oct-2003)

	Ref	Expression
Assertion	sdomentr	⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≈ 𝐶 ) → 𝐴 ≺ 𝐶 )

Proof

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