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Theorem disjdif 3296
 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 3157 . 2 (𝐴𝐵) ⊆ 𝐴
2 inssdif0im 3291 . 2 ((𝐴𝐵) ⊆ 𝐴 → (𝐴 ∩ (𝐵𝐴)) = ∅)
31, 2ax-mp 7 1 (𝐴 ∩ (𝐵𝐴)) = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∖ cdif 2914   ∩ cin 2916   ⊆ wss 2917  ∅c0 3224 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225 This theorem is referenced by:  ssdifin0  3304  difdifdirss  3307  fvsnun1  5360  fvsnun2  5361  phplem2  6316
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