Theorem exmidontriim 7571

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 2299	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
2		equequ1 1764	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
3		eleq2 2302	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
4	1, 2, 3	3orbi123d 1352	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)))
5	4	ralbidv 2550	. . . . 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 539	. . . . . . . 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 533	. . . . . . . 8 ⊢ ((((𝑥 ∈ On ∧ ∀𝑧 ∈ 𝑥 (EXMID → ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))) ∧ EXMID) ∧ 𝑎 ∈ On) → EXMID)
10		simpllr 540	. . . . . . . . 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 2618	. . . . . . . . . 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 493	. . . . . . . . 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 7570	. . . . . . 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 2623	. . . . . 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 2302	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → (𝑥 ∈ 𝑎 ↔ 𝑥 ∈ 𝑦))
18		equequ2 1765	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → (𝑥 = 𝑎 ↔ 𝑥 = 𝑦))
19		eleq1w 2299	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → (𝑎 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
20	17, 18, 19	3orbi123d 1352	. . . . . . 7 ⊢ (𝑎 = 𝑦 → ((𝑥 ∈ 𝑎 ∨ 𝑥 = 𝑎 ∨ 𝑎 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
21	20	cbvralv 2786	. . . . . 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 4727	. . 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 2623	1 ⊢ (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 104   ∨ w3o 1008   ∈ wcel 2209  ∀wral 2528  EXMIDwem 4326  Oncon0 4503

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-uni 3931  df-tr 4225  df-exmid 4327  df-iord 4506  df-on 4508

This theorem is referenced by:  exmidontri  7588  onntri51  7589  exmidontri2or  7592