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Theorem onirri 4515
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
onirri.1  |-  A  e.  On
Assertion
Ref Expression
onirri  |-  -.  A  e.  A

Proof of Theorem onirri
StepHypRef Expression
1 onirri.1 . . 3  |-  A  e.  On
21onordi 4399 . 2  |-  Ord  A
3 ordirr 4514 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
42, 3ax-mp 5 1  |-  -.  A  e.  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 2135   Ord word 4335   Oncon0 4336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146  ax-setind 4509
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-v 2724  df-dif 3114  df-in 3118  df-ss 3125  df-sn 3577  df-uni 3785  df-tr 4076  df-iord 4339  df-on 4341
This theorem is referenced by:  ontri2orexmidim  4544  enpr2d  6775  pm54.43  7138  pw1ne1  7177
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