Theorem ontri2orexmidim 4569

Description: Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4568. (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 4523	. . . . 5 ⊢ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
2		suc0 4409	. . . . . 6 ⊢ suc ∅ = {∅}
3		0elon 4390	. . . . . . 7 ⊢ ∅ ∈ On
4	3	onsuci 4513	. . . . . 6 ⊢ suc ∅ ∈ On
5	2, 4	eqeltrri 2251	. . . . 5 ⊢ {∅} ∈ On
6		sseq1 3178	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑥 ⊆ 𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦))
7		sseq2 3179	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
8	6, 7	orbi12d 793	. . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ∨ 𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
9		sseq2 3179	. . . . . . 7 ⊢ (𝑦 = {∅} → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅}))
10		sseq1 3178	. . . . . . 7 ⊢ (𝑦 = {∅} → (𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
11	9, 10	orbi12d 793	. . . . . 6 ⊢ (𝑦 = {∅} → (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ∨ 𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
12	8, 11	rspc2va 2855	. . . . 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 4540	. . . . . 6 ⊢ ¬ {∅} ∈ {∅}
15		simpl 109	. . . . . . . 8 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅})
16		simpr 110	. . . . . . . . 9 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → 𝜑)
17		p0ex 4186	. . . . . . . . . . 11 ⊢ {∅} ∈ V
18	17	prid2 3699	. . . . . . . . . 10 ⊢ {∅} ∈ {∅, {∅}}
19		biidd 172	. . . . . . . . . . 11 ⊢ (𝑧 = {∅} → (𝜑 ↔ 𝜑))
20	19	elrab3 2894	. . . . . . . . . 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 3156	. . . . . . 7 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {∅})
24	23	ex 115	. . . . . 6 ⊢ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → (𝜑 → {∅} ∈ {∅}))
25	14, 24	mtoi 664	. . . . 5 ⊢ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → ¬ 𝜑)
26		snssg 3726	. . . . . . 7 ⊢ (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
27	3, 26	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
28		0ex 4128	. . . . . . . 8 ⊢ ∅ ∈ V
29	28	prid1 3698	. . . . . . 7 ⊢ ∅ ∈ {∅, {∅}}
30		biidd 172	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
31	30	elrab3 2894	. . . . . . 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 759	. . . 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 729	. 2 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → (𝜑 ∨ ¬ 𝜑))
37		df-dc 835	. 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 708  DECID wdc 834   = wceq 1353   ∈ wcel 2148  ∀wral 2455  {crab 2459   ⊆ wss 3129  ∅c0 3422  {csn 3592  {cpr 3593  Oncon0 4361  suc csuc 4363

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 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  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 3577  df-sn 3598  df-pr 3599  df-uni 3809  df-tr 4100  df-iord 4364  df-on 4366  df-suc 4369

This theorem is referenced by:  exmidontri2or  7237