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Theorem onntri2or 7235
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 7233 . . 3 (¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥))
2 onntri24 7231 . . 3 (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥))
31, 2syl 14 . 2 (¬ ¬ EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥))
4 onntri45 7230 . 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 2453  wss 3127  EXMIDwem 4189  Oncon0 4357
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530
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 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-uni 3806  df-int 3841  df-tr 4097  df-exmid 4190  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-1o 6407  df-2o 6408  df-3o 6409
This theorem is referenced by: (None)
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