Theorem ontriexmidim 4664

Description: Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4663. (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 3525	. . . . . 6 ⊢ ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅
2	1	a1i 9	. . . . 5 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅)
3		ordtriexmidlem 4661	. . . . . . . 8 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
4		0elon 4532	. . . . . . . 8 ⊢ ∅ ∈ On
5		eleq1 2301	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦))
6		eqeq1 2245	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 = 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} = 𝑦))
7		eleq2 2302	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
8	5, 6, 7	3orbi123d 1352	. . . . . . . . 9 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ∨ {𝑧 ∈ {∅} ∣ 𝜑} = 𝑦 ∨ 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
9		eleq2 2302	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
10		eqeq2 2248	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜑} = 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅))
11		eleq1 2301	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
12	9, 10, 11	3orbi123d 1352	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ∨ {𝑧 ∈ {∅} ∣ 𝜑} = 𝑦 ∨ 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
13	8, 12	rspc2v 2943	. . . . . . . 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 430	. . . . . . 7 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
15		3orass 1012	. . . . . . 7 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
16	14, 15	sylib 122	. . . . . 6 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
17	16	orcomd 741	. . . . 5 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ∨ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
18	2, 17	ecased 1390	. . . 4 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
19		ordtriexmidlem2 4662	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
20		0ex 4255	. . . . . . . 8 ⊢ ∅ ∈ V
21	20	snid 3736	. . . . . . 7 ⊢ ∅ ∈ {∅}
22		biidd 172	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
23	22	elrab3 2983	. . . . . . 7 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
24	21, 23	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
25	24	biimpi 120	. . . . 5 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
26	19, 25	orim12i 771	. . . 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 741	. 2 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → (𝜑 ∨ ¬ 𝜑))
29		df-dc 847	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
30	28, 29	sylibr 134	1 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → DECID 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 720  DECID wdc 846   ∨ w3o 1008   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532  ∅c0 3520  {csn 3705  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

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  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-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511

This theorem is referenced by:  exmidontri  7588