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Theorem ssindif0 3983
 Description: Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
Assertion
Ref Expression
ssindif0 (𝐴𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Proof of Theorem ssindif0
StepHypRef Expression
1 disj2 3976 . 2 ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
2 ddif 3704 . . 3 (V ∖ (V ∖ 𝐵)) = 𝐵
32sseq2i 3593 . 2 (𝐴 ⊆ (V ∖ (V ∖ 𝐵)) ↔ 𝐴𝐵)
41, 3bitr2i 264 1 (𝐴𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  setind  8493
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