ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  onntri2or GIF version

Theorem onntri2or 7245
Description: Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
Assertion
Ref Expression
onntri2or (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥))
Distinct variable group:   𝑥,𝑦

Proof of Theorem onntri2or
StepHypRef Expression
1 onntri52 7243 . . 3 (¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥))
2 onntri24 7241 . . 3 (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥))
31, 2syl 14 . 2 (¬ ¬ EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥))
4 onntri45 7240 . 2 (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥) → ¬ ¬ EXMID)
53, 4impbii 126 1 (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 105  wo 708  wral 2455  wss 3130  EXMIDwem 4195  Oncon0 4364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-int 3846  df-tr 4103  df-exmid 4196  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-1o 6417  df-2o 6418  df-3o 6419
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator