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Theorem onirri 4629
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 4627 . 2  |-  Ord  A
3 ordirr 4541 . 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 1717   Ord word 4522   Oncon0 4523
This theorem is referenced by:  onssnel2i  4633  onuninsuci  4761  oelim2  6775  omopthlem2  6836  harndom  7466  wfelirr  7685  carduni  7802  pm54.43  7821  alephle  7903  alephfp  7923  pwxpndom2  8474  fvnobday  25361  onsucsuccmpi  25908  onint1  25914  wepwsolem  26808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527
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