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Theorem onirri 4821
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 4819 . 2  |-  Ord  A
3 ordirr 4733 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
42, 3ax-mp 5 1  |-  -.  A  e.  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1761   Ord word 4714   Oncon0 4715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-tr 4383  df-eprel 4628  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719
This theorem is referenced by:  onssnel2i  4825  onuninsuci  6450  oelim2  7030  omopthlem2  7091  harndom  7775  wfelirr  8028  carduni  8147  pm54.43  8166  alephle  8254  alephfp  8274  pwxpndom2  8828  fvnobday  27752  onsucsuccmpi  28219  onint1  28225  wepwsolem  29319
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