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Theorem ssindif0im 3390
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 3282 . . 3 𝐵 ⊆ (V ∖ (V ∖ 𝐵))
2 sstr 3073 . . 3 ((𝐴𝐵𝐵 ⊆ (V ∖ (V ∖ 𝐵))) → 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
31, 2mpan2 419 . 2 (𝐴𝐵𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
4 disj2 3386 . 2 ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
53, 4sylibr 133 1 (𝐴𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  Vcvv 2658  cdif 3036  cin 3038  wss 3039  c0 3331
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332
This theorem is referenced by: (None)
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