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

Theorem disjr 4357
 Description: Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
Assertion
Ref Expression
disjr ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem disjr
StepHypRef Expression
1 incom 4128 . . 3 (𝐴𝐵) = (𝐵𝐴)
21eqeq1i 2803 . 2 ((𝐴𝐵) = ∅ ↔ (𝐵𝐴) = ∅)
3 disj 4355 . 2 ((𝐵𝐴) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
42, 3bitri 278 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∩ cin 3880  ∅c0 4243 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rab 3115  df-dif 3884  df-in 3888  df-nul 4244 This theorem is referenced by:  kqdisj  22347  iccntr  23436  numedglnl  26947  fmlasucdisj  32774  ntrneicls11  40836  iooinlbub  42181  stoweidlem57  42742
 Copyright terms: Public domain W3C validator