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Theorem onirri 4499
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 4497 . 2  |-  Ord  A
3 ordirr 4410 . 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 1685   Ord word 4391   Oncon0 4392
This theorem is referenced by:  onssnel2i  4503  onuninsuci  4631  oelim2  6589  omopthlem2  6650  harndom  7274  carduni  7610  pm54.43  7629  alephle  7711  alephfp  7731  pwxpndom2  8283  axdenselem2  23738  rankeq1o  24209  onsucsuccmpi  24290  onint1  24296  wepwsolem  26538
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-tr 4116  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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