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Theorem ssindif0im 3281
 Description: Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
Assertion
Ref Expression
ssindif0im (𝐴𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Proof of Theorem ssindif0im
StepHypRef Expression
1 ddifss 3175 . . 3 𝐵 ⊆ (V ∖ (V ∖ 𝐵))
2 sstr 2953 . . 3 ((𝐴𝐵𝐵 ⊆ (V ∖ (V ∖ 𝐵))) → 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
31, 2mpan2 401 . 2 (𝐴𝐵𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
4 disj2 3275 . 2 ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
53, 4sylibr 137 1 (𝐴𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  Vcvv 2557   ∖ 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-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225 This theorem is referenced by: (None)
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