Theorem ordsoexmid 4539

Description: Weak linearity of ordinals implies the law of the excluded middle (that is, decidability of an arbitrary proposition). (Contributed by Mario Carneiro and Jim Kingdon, 29-Jan-2019.)

Hypothesis

Ref	Expression
ordsoexmid.1	⊢ E Or On

Assertion

Ref	Expression
ordsoexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Proof of Theorem ordsoexmid

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtriexmidlem 4496	. . . . 5 ⊢ {𝑤 ∈ {∅} ∣ 𝜑} ∈ On
2	1	elexi 2738	. . . 4 ⊢ {𝑤 ∈ {∅} ∣ 𝜑} ∈ V
3	2	sucid 4395	. . 3 ⊢ {𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}
4	1	onsuci 4493	. . . 4 ⊢ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On
5		suc0 4389	. . . . 5 ⊢ suc ∅ = {∅}
6		0elon 4370	. . . . . 6 ⊢ ∅ ∈ On
7	6	onsuci 4493	. . . . 5 ⊢ suc ∅ ∈ On
8	5, 7	eqeltrri 2240	. . . 4 ⊢ {∅} ∈ On
9		eleq1 2229	. . . . . . 7 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ On ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ On))
10	9	3anbi1d 1306	. . . . . 6 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On)))
11		eleq1 2229	. . . . . . 7 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
12		eleq1 2229	. . . . . . . 8 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅}))
13	12	orbi1d 781	. . . . . . 7 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
14	11, 13	imbi12d 233	. . . . . 6 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))))
15	10, 14	imbi12d 233	. . . . 5 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (((𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))) ↔ (({𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))))
16	4	elexi 2738	. . . . . 6 ⊢ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ V
17		eleq1 2229	. . . . . . . 8 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑦 ∈ On ↔ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On))
18	17	3anbi2d 1307	. . . . . . 7 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On) ↔ (𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On)))
19		eleq2 2230	. . . . . . . 8 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
20		eleq2 2230	. . . . . . . . 9 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ({∅} ∈ 𝑦 ↔ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
21	20	orbi2d 780	. . . . . . . 8 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦) ↔ (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
22	19, 21	imbi12d 233	. . . . . . 7 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ 𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)) ↔ (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))))
23	18, 22	imbi12d 233	. . . . . 6 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → (((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On) → (𝑥 ∈ 𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦))) ↔ ((𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))))
24		p0ex 4167	. . . . . . 7 ⊢ {∅} ∈ V
25		eleq1 2229	. . . . . . . . 9 ⊢ (𝑧 = {∅} → (𝑧 ∈ On ↔ {∅} ∈ On))
26	25	3anbi3d 1308	. . . . . . . 8 ⊢ (𝑧 = {∅} → ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑧 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On)))
27		eleq2 2230	. . . . . . . . . 10 ⊢ (𝑧 = {∅} → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ {∅}))
28		eleq1 2229	. . . . . . . . . 10 ⊢ (𝑧 = {∅} → (𝑧 ∈ 𝑦 ↔ {∅} ∈ 𝑦))
29	27, 28	orbi12d 783	. . . . . . . . 9 ⊢ (𝑧 = {∅} → ((𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦) ↔ (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)))
30	29	imbi2d 229	. . . . . . . 8 ⊢ (𝑧 = {∅} → ((𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)) ↔ (𝑥 ∈ 𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦))))
31	26, 30	imbi12d 233	. . . . . . 7 ⊢ (𝑧 = {∅} → (((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On) → (𝑥 ∈ 𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)))))
32		ordsoexmid.1	. . . . . . . . . . 11 ⊢ E Or On
33		df-iso 4275	. . . . . . . . . . 11 ⊢ ( E Or On ↔ ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧 ∨ 𝑧 E 𝑦))))
34	32, 33	mpbi 144	. . . . . . . . . 10 ⊢ ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧 ∨ 𝑧 E 𝑦)))
35	34	simpri 112	. . . . . . . . 9 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧 ∨ 𝑧 E 𝑦))
36		epel 4270	. . . . . . . . . . . 12 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
37		epel 4270	. . . . . . . . . . . . 13 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧)
38		epel 4270	. . . . . . . . . . . . 13 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
39	37, 38	orbi12i 754	. . . . . . . . . . . 12 ⊢ ((𝑥 E 𝑧 ∨ 𝑧 E 𝑦) ↔ (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))
40	36, 39	imbi12i 238	. . . . . . . . . . 11 ⊢ ((𝑥 E 𝑦 → (𝑥 E 𝑧 ∨ 𝑧 E 𝑦)) ↔ (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)))
41	40	2ralbii 2474	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧 ∨ 𝑧 E 𝑦)) ↔ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)))
42	41	ralbii 2472	. . . . . . . . 9 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧 ∨ 𝑧 E 𝑦)) ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)))
43	35, 42	mpbi 144	. . . . . . . 8 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))
44	43	rspec3 2556	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)))
45	24, 31, 44	vtocl 2780	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On) → (𝑥 ∈ 𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)))
46	16, 23, 45	vtocl 2780	. . . . 5 ⊢ ((𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
47	2, 15, 46	vtocl 2780	. . . 4 ⊢ (({𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
48	1, 4, 8, 47	mp3an 1327	. . 3 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
49	2	elsn 3592	. . . . 5 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ↔ {𝑤 ∈ {∅} ∣ 𝜑} = ∅)
50		ordtriexmidlem2 4497	. . . . 5 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
51	49, 50	sylbi 120	. . . 4 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
52		elirr 4518	. . . . . . 7 ⊢ ¬ {∅} ∈ {∅}
53		elrabi 2879	. . . . . . 7 ⊢ ({∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑} → {∅} ∈ {∅})
54	52, 53	mto 652	. . . . . 6 ⊢ ¬ {∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑}
55		elsuci 4381	. . . . . . 7 ⊢ ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑤 ∈ {∅} ∣ 𝜑}))
56	55	ord 714	. . . . . 6 ⊢ ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (¬ {∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑} → {∅} = {𝑤 ∈ {∅} ∣ 𝜑}))
57	54, 56	mpi 15	. . . . 5 ⊢ ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → {∅} = {𝑤 ∈ {∅} ∣ 𝜑})
58		0ex 4109	. . . . . . 7 ⊢ ∅ ∈ V
59		biidd 171	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝜑 ↔ 𝜑))
60	58, 59	rabsnt 3651	. . . . . 6 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} = {∅} → 𝜑)
61	60	eqcoms 2168	. . . . 5 ⊢ ({∅} = {𝑤 ∈ {∅} ∣ 𝜑} → 𝜑)
62	57, 61	syl 14	. . . 4 ⊢ ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → 𝜑)
63	51, 62	orim12i 749	. . 3 ⊢ (({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑))
64	3, 48, 63	mp2b 8	. 2 ⊢ (¬ 𝜑 ∨ 𝜑)
65		orcom 718	. 2 ⊢ ((¬ 𝜑 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
66	64, 65	mpbi 144	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ∀wral 2444  {crab 2448  ∅c0 3409  {csn 3576   class class class wbr 3982   E cep 4265   Po wpo 4272   Or wor 4273  Oncon0 4341  suc csuc 4343

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-tr 4081  df-eprel 4267  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by: (None)