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Theorem exmidontriim 7162
Description: Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.)
Assertion
Ref Expression
exmidontriim (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidontriim
Dummy variables 𝑎 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1w 2218 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝑦𝑧𝑦))
2 equequ1 1692 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
3 eleq2 2221 . . . . . . 7 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
41, 2, 33orbi123d 1293 . . . . . 6 (𝑥 = 𝑧 → ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑧𝑦𝑧 = 𝑦𝑦𝑧)))
54ralbidv 2457 . . . . 5 (𝑥 = 𝑧 → (∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧)))
65imbi2d 229 . . . 4 (𝑥 = 𝑧 → ((EXMID → ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥)) ↔ (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))))
7 simplll 523 . . . . . . . 8 ((((𝑥 ∈ On ∧ ∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → 𝑥 ∈ On)
8 simpr 109 . . . . . . . 8 ((((𝑥 ∈ On ∧ ∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → 𝑎 ∈ On)
9 simplr 520 . . . . . . . 8 ((((𝑥 ∈ On ∧ ∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → EXMID)
10 simpllr 524 . . . . . . . . 9 ((((𝑥 ∈ On ∧ ∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → ∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧)))
11 pm2.27 40 . . . . . . . . . . 11 (EXMID → ((EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧)) → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧)))
1211ralimdv 2525 . . . . . . . . . 10 (EXMID → (∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧)) → ∀𝑧𝑥𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧)))
1312ad2antlr 481 . . . . . . . . 9 ((((𝑥 ∈ On ∧ ∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → (∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧)) → ∀𝑧𝑥𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧)))
1410, 13mpd 13 . . . . . . . 8 ((((𝑥 ∈ On ∧ ∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → ∀𝑧𝑥𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))
157, 8, 9, 14exmidontriimlem4 7161 . . . . . . 7 ((((𝑥 ∈ On ∧ ∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → (𝑥𝑎𝑥 = 𝑎𝑎𝑥))
1615ralrimiva 2530 . . . . . 6 (((𝑥 ∈ On ∧ ∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))) ∧ EXMID) → ∀𝑎 ∈ On (𝑥𝑎𝑥 = 𝑎𝑎𝑥))
17 eleq2 2221 . . . . . . . 8 (𝑎 = 𝑦 → (𝑥𝑎𝑥𝑦))
18 equequ2 1693 . . . . . . . 8 (𝑎 = 𝑦 → (𝑥 = 𝑎𝑥 = 𝑦))
19 eleq1w 2218 . . . . . . . 8 (𝑎 = 𝑦 → (𝑎𝑥𝑦𝑥))
2017, 18, 193orbi123d 1293 . . . . . . 7 (𝑎 = 𝑦 → ((𝑥𝑎𝑥 = 𝑎𝑎𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
2120cbvralv 2680 . . . . . 6 (∀𝑎 ∈ On (𝑥𝑎𝑥 = 𝑎𝑎𝑥) ↔ ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
2216, 21sylib 121 . . . . 5 (((𝑥 ∈ On ∧ ∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))) ∧ EXMID) → ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
2322exp31 362 . . . 4 (𝑥 ∈ On → (∀𝑧𝑥 (EXMID → ∀𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧)) → (EXMID → ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥))))
246, 23tfis2 4546 . . 3 (𝑥 ∈ On → (EXMID → ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
2524impcom 124 . 2 ((EXMID𝑥 ∈ On) → ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
2625ralrimiva 2530 1 (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3o 962  wcel 2128  wral 2435  EXMIDwem 4157  Oncon0 4325
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-setind 4498
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-uni 3775  df-tr 4065  df-exmid 4158  df-iord 4328  df-on 4330
This theorem is referenced by:  exmidontri  7176  onntri51  7177  exmidontri2or  7180
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