Theorem exmidontriim 7285

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 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))

Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidontriim

Dummy variables 𝑎 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2254	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
2		equequ1 1723	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
3		eleq2 2257	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
4	1, 2, 3	3orbi123d 1322	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)))
5	4	ralbidv 2494	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)))
6	5	imbi2d 230	. . . 4 ⊢ (𝑥 = 𝑧 → ((EXMID → ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) ↔ (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))))
7		simplll 533	. . . . . . . 8 ⊢ ((((𝑥 ∈ On ∧ ∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → 𝑥 ∈ On)
8		simpr 110	. . . . . . . 8 ⊢ ((((𝑥 ∈ On ∧ ∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → 𝑎 ∈ On)
9		simplr 528	. . . . . . . 8 ⊢ ((((𝑥 ∈ On ∧ ∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → EXMID)
10		simpllr 534	. . . . . . . . 9 ⊢ ((((𝑥 ∈ On ∧ ∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → ∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)))
11		pm2.27 40	. . . . . . . . . . 11 ⊢ (EXMID → ((EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)) → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)))
12	11	ralimdv 2562	. . . . . . . . . 10 ⊢ (EXMID → (∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)) → ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)))
13	12	ad2antlr 489	. . . . . . . . 9 ⊢ ((((𝑥 ∈ On ∧ ∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → (∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)) → ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)))
14	10, 13	mpd 13	. . . . . . . 8 ⊢ ((((𝑥 ∈ On ∧ ∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))
15	7, 8, 9, 14	exmidontriimlem4 7284	. . . . . . 7 ⊢ ((((𝑥 ∈ On ∧ ∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → (𝑥 ∈ 𝑎 ∨ 𝑥 = 𝑎 ∨ 𝑎 ∈ 𝑥))
16	15	ralrimiva 2567	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ ∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))) ∧ EXMID) → ∀𝑎 ∈ On (𝑥 ∈ 𝑎 ∨ 𝑥 = 𝑎 ∨ 𝑎 ∈ 𝑥))
17		eleq2 2257	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → (𝑥 ∈ 𝑎 ↔ 𝑥 ∈ 𝑦))
18		equequ2 1724	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → (𝑥 = 𝑎 ↔ 𝑥 = 𝑦))
19		eleq1w 2254	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → (𝑎 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
20	17, 18, 19	3orbi123d 1322	. . . . . . 7 ⊢ (𝑎 = 𝑦 → ((𝑥 ∈ 𝑎 ∨ 𝑥 = 𝑎 ∨ 𝑎 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
21	20	cbvralv 2726	. . . . . 6 ⊢ (∀𝑎 ∈ On (𝑥 ∈ 𝑎 ∨ 𝑥 = 𝑎 ∨ 𝑎 ∈ 𝑥) ↔ ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
22	16, 21	sylib 122	. . . . 5 ⊢ (((𝑥 ∈ On ∧ ∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))) ∧ EXMID) → ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
23	22	exp31 364	. . . 4 ⊢ (𝑥 ∈ On → (∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)) → (EXMID → ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))))
24	6, 23	tfis2 4617	. . 3 ⊢ (𝑥 ∈ On → (EXMID → ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
25	24	impcom 125	. 2 ⊢ ((EXMID ∧ 𝑥 ∈ On) → ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
26	25	ralrimiva 2567	1 ⊢ (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 104   ∨ w3o 979   ∈ wcel 2164  ∀wral 2472  EXMIDwem 4223  Oncon0 4394

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-uni 3836  df-tr 4128  df-exmid 4224  df-iord 4397  df-on 4399

This theorem is referenced by:  exmidontri  7299  onntri51  7300  exmidontri2or  7303