Theorem ineqcomi

Description: Two ways of expressing that two classes have a given intersection. Inference form of ineqcom . Disjointness inference when C = (/) . (Contributed by Peter Mazsa, 26-Mar-2017) (Proof shortened by SN, 20-Sep-2024)

	Ref	Expression
Hypothesis	ineqcomi.1	⊢ ( 𝐴 ∩ 𝐵 ) = 𝐶
Assertion	ineqcomi	⊢ ( 𝐵 ∩ 𝐴 ) = 𝐶

Proof

Step	Hyp	Ref	Expression
1		ineqcomi.1	⊢ ( 𝐴 ∩ 𝐵 ) = 𝐶
2		incom	⊢ ( 𝐵 ∩ 𝐴 ) = ( 𝐴 ∩ 𝐵 )
3	2 1	eqtri	⊢ ( 𝐵 ∩ 𝐴 ) = 𝐶