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Theorem difin0 4394
 Description: The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difin0 ((𝐴𝐵) ∖ 𝐵) = ∅

Proof of Theorem difin0
StepHypRef Expression
1 inss2 4180 . 2 (𝐴𝐵) ⊆ 𝐵
2 ssdif0 4295 . 2 ((𝐴𝐵) ⊆ 𝐵 ↔ ((𝐴𝐵) ∖ 𝐵) = ∅)
31, 2mpbi 233 1 ((𝐴𝐵) ∖ 𝐵) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∖ cdif 3905   ∩ cin 3907   ⊆ wss 3908  ∅c0 4265 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-8 2116  ax-9 2124  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-dif 3911  df-in 3915  df-ss 3925  df-nul 4266 This theorem is referenced by:  volinun  24148
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