Theorem ensymb

Description: Symmetry of equinumerosity. Theorem 2 of Suppes p. 92. (Contributed by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Assertion	ensymb	⊢ ( 𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴 )

Proof

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