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Theorem ssindif0 4175
 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 4168 . 2 ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
2 ddif 3885 . . 3 (V ∖ (V ∖ 𝐵)) = 𝐵
32sseq2i 3771 . 2 (𝐴 ⊆ (V ∖ (V ∖ 𝐵)) ↔ 𝐴𝐵)
41, 3bitr2i 265 1 (𝐴𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1632  Vcvv 3340   ∖ cdif 3712   ∩ cin 3714   ⊆ wss 3715  ∅c0 4058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342  df-dif 3718  df-in 3722  df-ss 3729  df-nul 4059 This theorem is referenced by:  setind  8783
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