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Theorem disjdif 3510
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 3370 . 2 (𝐴𝐵) ⊆ 𝐴
2 inssdif0im 3505 . 2 ((𝐴𝐵) ⊆ 𝐴 → (𝐴 ∩ (𝐵𝐴)) = ∅)
31, 2ax-mp 5 1 (𝐴 ∩ (𝐵𝐴)) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1364  cdif 3141  cin 3143  wss 3144  c0 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438
This theorem is referenced by:  ssdifin0  3519  difdifdirss  3522  fvsnun1  5733  fvsnun2  5734  phplem2  6880  unfiin  6953  xpfi  6957  sbthlem7  6991  sbthlemi8  6992  exmidfodomrlemim  7229  fihashssdif  10829  zfz1isolem1  10851  fsumlessfi  11499  fprodsplit1f  11673  setsfun  12546  setsfun0  12547  setsslid  12562
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