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Theorem ordirr 4410
Description: Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. We prove this without invoking the Axiom of Regularity. (Contributed by NM, 2-Jan-1994.)
Assertion
Ref Expression
ordirr  |-  ( Ord 
A  ->  -.  A  e.  A )

Proof of Theorem ordirr
StepHypRef Expression
1 ordfr 4407 . 2  |-  ( Ord 
A  ->  _E  Fr  A )
2 efrirr 4374 . 2  |-  (  _E  Fr  A  ->  -.  A  e.  A )
31, 2syl 15 1  |-  ( Ord 
A  ->  -.  A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1684    _E cep 4303    Fr wfr 4349   Ord word 4391
This theorem is referenced by:  nordeq  4411  ordn2lp  4412  ordtri3or  4424  ordtri1  4425  ordtri3  4428  orddisj  4430  ordunidif  4440  ordnbtwn  4483  onirri  4499  onssneli  4502  onprc  4576  nlimsucg  4633  nnlim  4669  limom  4671  smo11  6381  smoord  6382  tfrlem13  6406  omopth2  6582  limensuci  7037  infensuc  7039  ordtypelem9  7241  cantnfp1lem3  7382  cantnfp1  7383  oemapvali  7386  wfelirr  7497  tskwe  7583  dif1card  7638  pm110.643ALT  7804  pwsdompw  7830  cflim2  7889  fin23lem24  7948  fin23lem26  7951  axdc3lem4  8079  ttukeylem7  8142  canthp1lem2  8275  inar1  8397  gruina  8440  grur1  8442  addnidpi  8525  fzennn  11030  hashp1i  11369  soseq  24254  hfninf  24816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-fr 4352  df-we 4354  df-ord 4395
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