Theorem ineqcom

Description: Two ways of expressing that two classes have a given intersection. This is often used when that given intersection is the empty set, in which case the statement displays two ways of expressing that two classes are disjoint (when C = (/) : ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) ) ). (Contributed by Peter Mazsa, 22-Mar-2017)

	Ref	Expression
Assertion	ineqcom	⊢ ( ( 𝐴 ∩ 𝐵 ) = 𝐶 ↔ ( 𝐵 ∩ 𝐴 ) = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		incom	⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 )
2	1	eqeq1i	⊢ ( ( 𝐴 ∩ 𝐵 ) = 𝐶 ↔ ( 𝐵 ∩ 𝐴 ) = 𝐶 )