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Theorem onirri 4651
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 4649 . 2  |-  Ord  A
3 ordirr 4563 . 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 1721   Ord word 4544   Oncon0 4545
This theorem is referenced by:  onssnel2i  4655  onuninsuci  4783  oelim2  6801  omopthlem2  6862  harndom  7492  wfelirr  7711  carduni  7828  pm54.43  7847  alephle  7929  alephfp  7949  pwxpndom2  8500  fvnobday  25554  onsucsuccmpi  26101  onint1  26107  wepwsolem  27010
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-tr 4267  df-eprel 4458  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549
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