Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjne Structured version   Unicode version

Theorem disjne 3675
 Description: Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
disjne

Proof of Theorem disjne
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 disj 3670 . . 3
2 eleq1 2498 . . . . . 6
32notbid 287 . . . . 5
43rspccva 3053 . . . 4
5 eleq1a 2507 . . . . 5
65necon3bd 2640 . . . 4
74, 6syl5com 29 . . 3
81, 7sylanb 460 . 2
983impia 1151 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726   wne 2601  wral 2707   cin 3321  c0 3630 This theorem is referenced by:  brdom7disj  8411  brdom6disj  8412  kelac1  27140  frlmssuvc1  27225  frlmsslsp  27227 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-v 2960  df-dif 3325  df-in 3329  df-nul 3631
 Copyright terms: Public domain W3C validator