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Theorem ordirr 4559
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 4556 . 2  |-  ( Ord 
A  ->  _E  Fr  A )
2 efrirr 4523 . 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 1721    _E cep 4452    Fr wfr 4498   Ord word 4540
This theorem is referenced by:  nordeq  4560  ordn2lp  4561  ordtri3or  4573  ordtri1  4574  ordtri3  4577  orddisj  4579  ordunidif  4589  ordnbtwn  4631  onirri  4647  onssneli  4650  onprc  4724  nlimsucg  4781  nnlim  4817  limom  4819  smo11  6585  smoord  6586  tfrlem13  6610  omopth2  6786  limensuci  7242  infensuc  7244  ordtypelem9  7451  cantnfp1lem3  7592  cantnfp1  7593  oemapvali  7596  tskwe  7793  dif1card  7848  pm110.643ALT  8014  pwsdompw  8040  cflim2  8099  fin23lem24  8158  fin23lem26  8161  axdc3lem4  8289  ttukeylem7  8351  canthp1lem2  8484  inar1  8606  gruina  8649  grur1  8651  addnidpi  8734  fzennn  11262  hashp1i  11627  soseq  25468
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-eprel 4454  df-fr 4501  df-we 4503  df-ord 4544
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