 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssindif0 Structured version   Visualization version   GIF version

Theorem ssindif0 4064
 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 4057 . 2 ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
2 ddif 3775 . . 3 (V ∖ (V ∖ 𝐵)) = 𝐵
32sseq2i 3663 . 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 1523  Vcvv 3231   ∖ cdif 3604   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949 This theorem is referenced by:  setind  8648
 Copyright terms: Public domain W3C validator