Theorem ontriexmidim 4504

Description: Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4503. (Contributed by Jim Kingdon, 26-Aug-2024.)

Assertion

Ref	Expression
ontriexmidim	⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → DECID 𝜑)

Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ontriexmidim

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3418	. . . . . 6 ⊢ ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅
2	1	a1i 9	. . . . 5 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅)
3		ordtriexmidlem 4501	. . . . . . . 8 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
4		0elon 4375	. . . . . . . 8 ⊢ ∅ ∈ On
5		eleq1 2233	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦))
6		eqeq1 2177	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 = 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} = 𝑦))
7		eleq2 2234	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
8	5, 6, 7	3orbi123d 1306	. . . . . . . . 9 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ∨ {𝑧 ∈ {∅} ∣ 𝜑} = 𝑦 ∨ 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
9		eleq2 2234	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
10		eqeq2 2180	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜑} = 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅))
11		eleq1 2233	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
12	9, 10, 11	3orbi123d 1306	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ∨ {𝑧 ∈ {∅} ∣ 𝜑} = 𝑦 ∨ 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
13	8, 12	rspc2v 2847	. . . . . . . 8 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ∈ On ∧ ∅ ∈ On) → (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
14	3, 4, 13	mp2an 424	. . . . . . 7 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
15		3orass 976	. . . . . . 7 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
16	14, 15	sylib 121	. . . . . 6 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
17	16	orcomd 724	. . . . 5 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ∨ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
18	2, 17	ecased 1344	. . . 4 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
19		ordtriexmidlem2 4502	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
20		0ex 4114	. . . . . . . 8 ⊢ ∅ ∈ V
21	20	snid 3612	. . . . . . 7 ⊢ ∅ ∈ {∅}
22		biidd 171	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
23	22	elrab3 2887	. . . . . . 7 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
24	21, 23	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
25	24	biimpi 119	. . . . 5 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
26	19, 25	orim12i 754	. . . 4 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑))
27	18, 26	syl 14	. . 3 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → (¬ 𝜑 ∨ 𝜑))
28	27	orcomd 724	. 2 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → (𝜑 ∨ ¬ 𝜑))
29		df-dc 830	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
30	28, 29	sylibr 133	1 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → DECID 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 703  DECID wdc 829   ∨ w3o 972   = wceq 1348   ∈ wcel 2141  ∀wral 2448  {crab 2452  ∅c0 3414  {csn 3581  Oncon0 4346

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-uni 3795  df-tr 4086  df-iord 4349  df-on 4351  df-suc 4354

This theorem is referenced by:  exmidontri  7203