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Theorem ineqcom 34326
 Description: Two ways of saying that two classes are disjoint (when 𝐶 = ∅: ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)). (Contributed by Peter Mazsa, 22-Mar-2017.)
Assertion
Ref Expression
ineqcom ((𝐴𝐵) = 𝐶 ↔ (𝐵𝐴) = 𝐶)

Proof of Theorem ineqcom
StepHypRef Expression
1 incom 4004 . 2 (𝐴𝐵) = (𝐵𝐴)
21eqeq1i 2811 1 ((𝐴𝐵) = 𝐶 ↔ (𝐵𝐴) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 197   = wceq 1637   ∩ cin 3768 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784 This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393  df-in 3776 This theorem is referenced by:  ineqcomi  34327
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