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Theorem ordirr 4533
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 4530 . 2  |-  ( Ord 
A  ->  _E  Fr  A )
2 efrirr 4497 . 2  |-  (  _E  Fr  A  ->  -.  A  e.  A )
31, 2syl 16 1  |-  ( Ord 
A  ->  -.  A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1717    _E cep 4426    Fr wfr 4472   Ord word 4514
This theorem is referenced by:  nordeq  4534  ordn2lp  4535  ordtri3or  4547  ordtri1  4548  ordtri3  4551  orddisj  4553  ordunidif  4563  ordnbtwn  4605  onirri  4621  onssneli  4624  onprc  4698  nlimsucg  4755  nnlim  4791  limom  4793  smo11  6555  smoord  6556  tfrlem13  6580  omopth2  6756  limensuci  7212  infensuc  7214  ordtypelem9  7421  cantnfp1lem3  7562  cantnfp1  7563  oemapvali  7566  tskwe  7763  dif1card  7818  pm110.643ALT  7984  pwsdompw  8010  cflim2  8069  fin23lem24  8128  fin23lem26  8131  axdc3lem4  8259  ttukeylem7  8321  canthp1lem2  8454  inar1  8576  gruina  8619  grur1  8621  addnidpi  8704  fzennn  11227  hashp1i  11592  soseq  25271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-eprel 4428  df-fr 4475  df-we 4477  df-ord 4518
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