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

Theorem onirri 4515
Description: An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
Hypothesis
Ref Expression
on.1  |-  A  e.  On
Assertion
Ref Expression
onirri  |-  -.  A  e.  A

Proof of Theorem onirri
StepHypRef Expression
1 on.1 . . 3  |-  A  e.  On
21onordi 4513 . 2  |-  Ord  A
3 ordirr 4426 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
42, 3ax-mp 8 1  |-  -.  A  e.  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1696   Ord word 4407   Oncon0 4408
This theorem is referenced by:  onssnel2i  4519  onuninsuci  4647  oelim2  6609  omopthlem2  6670  harndom  7294  carduni  7630  pm54.43  7649  alephle  7731  alephfp  7751  pwxpndom2  8303  fvnobday  24407  rankeq1o  24873  onsucsuccmpi  24954  onint1  24960  wepwsolem  27241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
  Copyright terms: Public domain W3C validator