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Theorem onntri3or 7237
Description: Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
Assertion
Ref Expression
onntri3or (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
Distinct variable group:   𝑥,𝑦

Proof of Theorem onntri3or
StepHypRef Expression
1 onntri51 7232 . . 3 (¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
2 onntri13 7230 . . 3 (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
31, 2syl 14 . 2 (¬ ¬ EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
4 onntri35 7229 . 2 (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ¬ ¬ EXMID)
53, 4impbii 126 1 (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 105  w3o 977  wral 2455  EXMIDwem 4191  Oncon0 4359
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 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
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 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-uni 3808  df-int 3843  df-tr 4099  df-exmid 4192  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-1o 6410  df-2o 6411  df-3o 6412
This theorem is referenced by: (None)
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