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Theorem onirri 4471
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 4469 . 2  |-  Ord  A
3 ordirr 4382 . 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 4363   Oncon0 4364
This theorem is referenced by:  onssnel2i  4475  onuninsuci  4603  oelim2  6561  omopthlem2  6622  harndom  7246  carduni  7582  pm54.43  7601  alephle  7683  alephfp  7703  pwxpndom2  8255  axdenselem2  23705  rankeq1o  24176  onsucsuccmpi  24257  onint1  24263  wepwsolem  26505
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368
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