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Theorem ordirr 4896
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 4893 . 2  |-  ( Ord 
A  ->  _E  Fr  A )
2 efrirr 4860 . 2  |-  (  _E  Fr  A  ->  -.  A  e.  A )
31, 2syl 16 1  |-  ( Ord 
A  ->  -.  A  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1767    _E cep 4789    Fr wfr 4835   Ord word 4877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-eprel 4791  df-fr 4838  df-we 4840  df-ord 4881
This theorem is referenced by:  nordeq  4897  ordn2lp  4898  ordtri3or  4910  ordtri1  4911  ordtri3  4914  orddisj  4916  ordunidif  4926  ordnbtwn  4968  onirri  4984  onssneli  4987  onprc  6604  nlimsucg  6661  nnlim  6697  limom  6699  smo11  7035  smoord  7036  tfrlem13  7059  omopth2  7233  limensuci  7693  infensuc  7695  ordtypelem9  7951  cantnfp1lem3  8099  cantnfp1  8100  oemapvali  8103  cantnfp1lem3OLD  8125  cantnfp1OLD  8126  tskwe  8331  dif1card  8388  pm110.643ALT  8558  pwsdompw  8584  cflim2  8643  fin23lem24  8702  fin23lem26  8705  axdc3lem4  8833  ttukeylem7  8895  canthp1lem2  9031  inar1  9153  gruina  9196  grur1  9198  addnidpi  9279  fzennn  12046  hashp1i  12434  soseq  28939
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