Theorem ontri2orexmidim 4604

Description: Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4603. (Contributed by Jim Kingdon, 26-Aug-2024.)

Assertion

Ref	Expression
ontri2orexmidim	⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → DECID 𝜑)

Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ontri2orexmidim

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtri2or2exmidlem 4558	. . . . 5 ⊢ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
2		suc0 4442	. . . . . 6 ⊢ suc ∅ = {∅}
3		0elon 4423	. . . . . . 7 ⊢ ∅ ∈ On
4	3	onsuci 4548	. . . . . 6 ⊢ suc ∅ ∈ On
5	2, 4	eqeltrri 2267	. . . . 5 ⊢ {∅} ∈ On
6		sseq1 3202	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑥 ⊆ 𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦))
7		sseq2 3203	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
8	6, 7	orbi12d 794	. . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ∨ 𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
9		sseq2 3203	. . . . . . 7 ⊢ (𝑦 = {∅} → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅}))
10		sseq1 3202	. . . . . . 7 ⊢ (𝑦 = {∅} → (𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
11	9, 10	orbi12d 794	. . . . . 6 ⊢ (𝑦 = {∅} → (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ∨ 𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
12	8, 11	rspc2va 2878	. . . . 5 ⊢ ((({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
13	1, 5, 12	mpanl12 436	. . . 4 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
14	5	onirri 4575	. . . . . 6 ⊢ ¬ {∅} ∈ {∅}
15		simpl 109	. . . . . . . 8 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅})
16		simpr 110	. . . . . . . . 9 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → 𝜑)
17		p0ex 4217	. . . . . . . . . . 11 ⊢ {∅} ∈ V
18	17	prid2 3725	. . . . . . . . . 10 ⊢ {∅} ∈ {∅, {∅}}
19		biidd 172	. . . . . . . . . . 11 ⊢ (𝑧 = {∅} → (𝜑 ↔ 𝜑))
20	19	elrab3 2917	. . . . . . . . . 10 ⊢ ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
21	18, 20	ax-mp 5	. . . . . . . . 9 ⊢ ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
22	16, 21	sylibr 134	. . . . . . . 8 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
23	15, 22	sseldd 3180	. . . . . . 7 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {∅})
24	23	ex 115	. . . . . 6 ⊢ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → (𝜑 → {∅} ∈ {∅}))
25	14, 24	mtoi 665	. . . . 5 ⊢ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → ¬ 𝜑)
26		snssg 3752	. . . . . . 7 ⊢ (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
27	3, 26	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
28		0ex 4156	. . . . . . . 8 ⊢ ∅ ∈ V
29	28	prid1 3724	. . . . . . 7 ⊢ ∅ ∈ {∅, {∅}}
30		biidd 172	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
31	30	elrab3 2917	. . . . . . 7 ⊢ (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
32	29, 31	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
33	27, 32	sylbb1 137	. . . . 5 ⊢ ({∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
34	25, 33	orim12i 760	. . . 4 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑))
35	13, 34	syl 14	. . 3 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → (¬ 𝜑 ∨ 𝜑))
36	35	orcomd 730	. 2 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → (𝜑 ∨ ¬ 𝜑))
37		df-dc 836	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
38	36, 37	sylibr 134	1 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → DECID 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2164  ∀wral 2472  {crab 2476   ⊆ wss 3153  ∅c0 3446  {csn 3618  {cpr 3619  Oncon0 4394  suc csuc 4396

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-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402

This theorem is referenced by:  exmidontri2or  7303