ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordirr GIF version

Theorem ordirr 4425
Description: Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. The present proof requires ax-setind 4420. If in the definition of ordinals df-iord 4256, we also required that membership be well-founded on any ordinal (see df-frind 4222), then we could prove ordirr 4425 without ax-setind 4420. (Contributed by NM, 2-Jan-1994.)
Assertion
Ref Expression
ordirr (Ord 𝐴 → ¬ 𝐴𝐴)

Proof of Theorem ordirr
StepHypRef Expression
1 elirr 4424 . 2 ¬ 𝐴𝐴
21a1i 9 1 (Ord 𝐴 → ¬ 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1463  Ord word 4252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-v 2660  df-dif 3041  df-sn 3501
This theorem is referenced by:  onirri  4426  nordeq  4427  ordn2lp  4428  orddisj  4429  onprc  4435  nlimsucg  4449  tfr1onlemsucfn  6203  tfr1onlemsucaccv  6204  tfrcllemsucfn  6216  tfrcllemsucaccv  6217  nntr2  6365  unsnfi  6773  addnidpig  7108  frecfzennn  10139  hashinfom  10464  hashennn  10466  hashp1i  10496  ennnfonelemg  11811  ctinfom  11836  nninfalllemn  13013
  Copyright terms: Public domain W3C validator