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Theorem ordirr 4513
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 4510 . 2  |-  ( Ord 
A  ->  _E  Fr  A )
2 efrirr 4477 . 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 1715    _E cep 4406    Fr wfr 4452   Ord word 4494
This theorem is referenced by:  nordeq  4514  ordn2lp  4515  ordtri3or  4527  ordtri1  4528  ordtri3  4531  orddisj  4533  ordunidif  4543  ordnbtwn  4586  onirri  4602  onssneli  4605  onprc  4679  nlimsucg  4736  nnlim  4772  limom  4774  smo11  6523  smoord  6524  tfrlem13  6548  omopth2  6724  limensuci  7180  infensuc  7182  ordtypelem9  7388  cantnfp1lem3  7529  cantnfp1  7530  oemapvali  7533  wfelirr  7644  tskwe  7730  dif1card  7785  pm110.643ALT  7951  pwsdompw  7977  cflim2  8036  fin23lem24  8095  fin23lem26  8098  axdc3lem4  8226  ttukeylem7  8289  canthp1lem2  8422  inar1  8544  gruina  8587  grur1  8589  addnidpi  8672  fzennn  11194  hashp1i  11559  soseq  25080  hfninf  25643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-eprel 4408  df-fr 4455  df-we 4457  df-ord 4498
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