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Theorem disjdif 3439
 Description: A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
Assertion
Ref Expression
disjdif (𝐴 ∩ (𝐵𝐴)) = ∅

Proof of Theorem disjdif
StepHypRef Expression
1 inss1 3300 . 2 (𝐴𝐵) ⊆ 𝐴
2 inssdif0im 3434 . 2 ((𝐴𝐵) ⊆ 𝐴 → (𝐴 ∩ (𝐵𝐴)) = ∅)
31, 2ax-mp 5 1 (𝐴 ∩ (𝐵𝐴)) = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1332   ∖ cdif 3072   ∩ cin 3074   ⊆ wss 3075  ∅c0 3367 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3077  df-in 3081  df-ss 3088  df-nul 3368 This theorem is referenced by:  ssdifin0  3448  difdifdirss  3451  fvsnun1  5624  fvsnun2  5625  phplem2  6754  unfiin  6821  xpfi  6825  sbthlem7  6858  sbthlemi8  6859  exmidfodomrlemim  7073  fihashssdif  10595  zfz1isolem1  10614  fsumlessfi  11260  setsfun  12031  setsfun0  12032  setsslid  12046
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