Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  disjr Structured version   GIF version

Theorem disjr 3263
 Description: Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
Assertion
Ref Expression
disjr ((AB) = ∅ ↔ x B ¬ x A)
Distinct variable groups:   x,A   x,B

Proof of Theorem disjr
StepHypRef Expression
1 incom 3123 . . 3 (AB) = (BA)
21eqeq1i 2044 . 2 ((AB) = ∅ ↔ (BA) = ∅)
3 disj 3262 . 2 ((BA) = ∅ ↔ x B ¬ x A)
42, 3bitri 173 1 ((AB) = ∅ ↔ x B ¬ x A)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300   ∩ cin 2910  ∅c0 3218 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-nul 3219 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator