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Theorem ineqcom 35704
 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 4128 . 2 (𝐴𝐵) = (𝐵𝐴)
21eqeq1i 2803 1 ((𝐴𝐵) = 𝐶 ↔ (𝐵𝐴) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∩ cin 3880 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-rab 3115  df-in 3888 This theorem is referenced by:  ineqcomi  35705
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