Theorem ineqcomi 4140

Description: Two ways of expressing that two classes have a given intersection. Inference form of ineqcom 4139. 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 4138	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		ineqcomi.1	. 2 ⊢ (𝐴 ∩ 𝐵) = 𝐶
3	1, 2	eqtri 2762	1 ⊢ (𝐵 ∩ 𝐴) = 𝐶

Syntax hints:   = wceq 1547   ∩ cin 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-rab 3392  df-in 3890

This theorem is referenced by:  dfss7  4179  0in  4325  disjdifr  4401  iinrab2  4999  cnvimainrn  7008  cnfldfunALT  21362  psdmul  22154  xrlimcnp  26950  nn0diffz0  32886  vonf1owev  35336  inres2  38614  ecqmap  38816  readvrec  42839  isubgr0uhgr  48364