Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nelne1 Structured version   Unicode version

Theorem nelne1 2695
 Description: Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
Assertion
Ref Expression
nelne1

Proof of Theorem nelne1
StepHypRef Expression
1 eleq2 2499 . . . 4
21biimpcd 217 . . 3
32necon3bd 2640 . 2
43imp 420 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360   wceq 1653   wcel 1726   wne 2601 This theorem is referenced by:  difsnb  3942  fofinf1o  7390  fin23lem24  8207  fin23lem31  8228  ttukeylem7  8400  npomex  8878  lbspss  16159  islbs3  16232  lbsextlem4  16238  obslbs  16962  hauspwpwf1  18024  ppiltx  20965  ex-pss  21741  cntnevol  24587  rpnnen3lem  27116  lshpnelb  29856  osumcllem10N  30836  pexmidlem7N  30847  dochsnkrlem1  32341 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-cleq 2431  df-clel 2434  df-ne 2603
 Copyright terms: Public domain W3C validator