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Theorem onirri 4390
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 4388 . 2  |-  Ord  A
3 ordirr 4303 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
42, 3ax-mp 10 1  |-  -.  A  e.  A
Colors of variables: wff set class
Syntax hints:   -. wn 5    e. wcel 1621   Ord word 4284   Oncon0 4285
This theorem is referenced by:  onssnel2i  4394  onuninsuci  4522  oelim2  6479  omopthlem2  6540  harndom  7162  carduni  7498  pm54.43  7517  alephle  7599  alephfp  7619  pwxpndom2  8167  axdenselem2  23504  rankeq1o  23975  onsucsuccmpi  24056  onint1  24062  wepwsolem  26304
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289
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