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Theorem disjdif 3323
 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 3193 . 2 (𝐴𝐵) ⊆ 𝐴
2 inssdif0im 3318 . 2 ((𝐴𝐵) ⊆ 𝐴 → (𝐴 ∩ (𝐵𝐴)) = ∅)
31, 2ax-mp 7 1 (𝐴 ∩ (𝐵𝐴)) = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1285   ∖ cdif 2971   ∩ cin 2973   ⊆ wss 2974  ∅c0 3258 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-in 2980  df-ss 2987  df-nul 3259 This theorem is referenced by:  ssdifin0  3331  difdifdirss  3334  fvsnun1  5392  fvsnun2  5393  phplem2  6388  unfiin  6444
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