Theorem ineqcomi 4161

Description: Two ways of expressing that two classes have a given intersection. Inference form of ineqcom 4160. Disjointness inference when 𝐶 = ∅. (Contributed by Peter Mazsa, 26-Mar-2017.) (Proof shortened by SN, 20-Sep-2024.)

Hypothesis

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

Assertion

Ref	Expression
ineqcomi	⊢ (𝐵 ∩ 𝐴) = 𝐶

Proof of Theorem ineqcomi

Step	Hyp	Ref	Expression
1		incom 4159	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		ineqcomi.1	. 2 ⊢ (𝐴 ∩ 𝐵) = 𝐶
3	1, 2	eqtri 2757	1 ⊢ (𝐵 ∩ 𝐴) = 𝐶

Syntax hints:   = wceq 1541   ∩ cin 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-rab 3398  df-in 3906

This theorem is referenced by:  cnvimainrn  7010  cnfldfunALT  21322  cnfldfunALTOLD  21335  psdmul  22107  xrlimcnp  26932  nn0diffz0  32823  vonf1owev  35251  inres2  38382  readvrec  42559  isubgr0uhgr  48061