ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordn2lp GIF version

Theorem ordn2lp 4374
Description: An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
ordn2lp (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))

Proof of Theorem ordn2lp
StepHypRef Expression
1 ordirr 4371 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
2 ordtr 4214 . . 3 (Ord 𝐴 → Tr 𝐴)
3 trel 3949 . . 3 (Tr 𝐴 → ((𝐴𝐵𝐵𝐴) → 𝐴𝐴))
42, 3syl 14 . 2 (Ord 𝐴 → ((𝐴𝐵𝐵𝐴) → 𝐴𝐴))
51, 4mtod 625 1 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wcel 1439  Tr wtr 3942  Ord word 4198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-setind 4366
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-v 2622  df-dif 3002  df-in 3006  df-ss 3013  df-sn 3456  df-uni 3660  df-tr 3943  df-iord 4202
This theorem is referenced by:  nninfalllemn  12170
  Copyright terms: Public domain W3C validator