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Theorem difin0 3992
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 3795 . 2 (𝐴𝐵) ⊆ 𝐵
2 ssdif0 3895 . 2 ((𝐴𝐵) ⊆ 𝐵 ↔ ((𝐴𝐵) ∖ 𝐵) = ∅)
31, 2mpbi 218 1 ((𝐴𝐵) ∖ 𝐵) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  cdif 3536  cin 3538  wss 3539  c0 3873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-dif 3542  df-in 3546  df-ss 3553  df-nul 3874
This theorem is referenced by:  volinun  23038
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