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

Theorem ssindif0 4470
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 4464 . 2 ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
2 ddif 4151 . . 3 (V ∖ (V ∖ 𝐵)) = 𝐵
32sseq2i 4025 . 2 (𝐴 ⊆ (V ∖ (V ∖ 𝐵)) ↔ 𝐴𝐵)
41, 3bitr2i 276 1 (𝐴𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  Vcvv 3478  cdif 3960  cin 3962  wss 3963  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340
This theorem is referenced by:  setind  9772
  Copyright terms: Public domain W3C validator