Theorem entr

Description: Transitivity of equinumerosity. Theorem 3 of Suppes p. 92. (Contributed by NM, 9-Jun-1998)

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

Proof

Step	Hyp	Ref	Expression
1		ener	⊢ ≈ Er V
2	1	a1i	⊢ ( ⊤ → ≈ Er V )
3	2	ertr	⊢ ( ⊤ → ( ( 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶 ) → 𝐴 ≈ 𝐶 ) )
4	3	mptru	⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶 ) → 𝐴 ≈ 𝐶 )