Theorem ordsoexmid 4240

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 4208	. . . . 5 ⊢ {w ∈ {∅} ∣ φ} ∈ On
2	1	elexi 2561	. . . 4 ⊢ {w ∈ {∅} ∣ φ} ∈ V
3	2	sucid 4120	. . 3 ⊢ {w ∈ {∅} ∣ φ} ∈ suc {w ∈ {∅} ∣ φ}
4	1	onsuci 4207	. . . 4 ⊢ suc {w ∈ {∅} ∣ φ} ∈ On
5		suc0 4114	. . . . 5 ⊢ suc ∅ = {∅}
6		0elon 4095	. . . . . 6 ⊢ ∅ ∈ On
7	6	onsuci 4207	. . . . 5 ⊢ suc ∅ ∈ On
8	5, 7	eqeltrri 2108	. . . 4 ⊢ {∅} ∈ On
9		eleq1 2097	. . . . . . 7 ⊢ (x = {w ∈ {∅} ∣ φ} → (x ∈ On ↔ {w ∈ {∅} ∣ φ} ∈ On))
10	9	3anbi1d 1210	. . . . . 6 ⊢ (x = {w ∈ {∅} ∣ φ} → ((x ∈ On ∧ suc {w ∈ {∅} ∣ φ} ∈ On ∧ {∅} ∈ On) ↔ ({w ∈ {∅} ∣ φ} ∈ On ∧ suc {w ∈ {∅} ∣ φ} ∈ On ∧ {∅} ∈ On)))
11		eleq1 2097	. . . . . . 7 ⊢ (x = {w ∈ {∅} ∣ φ} → (x ∈ suc {w ∈ {∅} ∣ φ} ↔ {w ∈ {∅} ∣ φ} ∈ suc {w ∈ {∅} ∣ φ}))
12		eleq1 2097	. . . . . . . 8 ⊢ (x = {w ∈ {∅} ∣ φ} → (x ∈ {∅} ↔ {w ∈ {∅} ∣ φ} ∈ {∅}))
13	12	orbi1d 704	. . . . . . 7 ⊢ (x = {w ∈ {∅} ∣ φ} → ((x ∈ {∅} ∨ {∅} ∈ suc {w ∈ {∅} ∣ φ}) ↔ ({w ∈ {∅} ∣ φ} ∈ {∅} ∨ {∅} ∈ suc {w ∈ {∅} ∣ φ})))
14	11, 13	imbi12d 223	. . . . . 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 223	. . . . 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 2561	. . . . . 6 ⊢ suc {w ∈ {∅} ∣ φ} ∈ V
17		eleq1 2097	. . . . . . . 8 ⊢ (y = suc {w ∈ {∅} ∣ φ} → (y ∈ On ↔ suc {w ∈ {∅} ∣ φ} ∈ On))
18	17	3anbi2d 1211	. . . . . . 7 ⊢ (y = suc {w ∈ {∅} ∣ φ} → ((x ∈ On ∧ y ∈ On ∧ {∅} ∈ On) ↔ (x ∈ On ∧ suc {w ∈ {∅} ∣ φ} ∈ On ∧ {∅} ∈ On)))
19		eleq2 2098	. . . . . . . 8 ⊢ (y = suc {w ∈ {∅} ∣ φ} → (x ∈ y ↔ x ∈ suc {w ∈ {∅} ∣ φ}))
20		eleq2 2098	. . . . . . . . 9 ⊢ (y = suc {w ∈ {∅} ∣ φ} → ({∅} ∈ y ↔ {∅} ∈ suc {w ∈ {∅} ∣ φ}))
21	20	orbi2d 703	. . . . . . . 8 ⊢ (y = suc {w ∈ {∅} ∣ φ} → ((x ∈ {∅} ∨ {∅} ∈ y) ↔ (x ∈ {∅} ∨ {∅} ∈ suc {w ∈ {∅} ∣ φ})))
22	19, 21	imbi12d 223	. . . . . . 7 ⊢ (y = suc {w ∈ {∅} ∣ φ} → ((x ∈ y → (x ∈ {∅} ∨ {∅} ∈ y)) ↔ (x ∈ suc {w ∈ {∅} ∣ φ} → (x ∈ {∅} ∨ {∅} ∈ suc {w ∈ {∅} ∣ φ}))))
23	18, 22	imbi12d 223	. . . . . 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 3930	. . . . . . 7 ⊢ {∅} ∈ V
25		eleq1 2097	. . . . . . . . 9 ⊢ (z = {∅} → (z ∈ On ↔ {∅} ∈ On))
26	25	3anbi3d 1212	. . . . . . . 8 ⊢ (z = {∅} → ((x ∈ On ∧ y ∈ On ∧ z ∈ On) ↔ (x ∈ On ∧ y ∈ On ∧ {∅} ∈ On)))
27		eleq2 2098	. . . . . . . . . 10 ⊢ (z = {∅} → (x ∈ z ↔ x ∈ {∅}))
28		eleq1 2097	. . . . . . . . . 10 ⊢ (z = {∅} → (z ∈ y ↔ {∅} ∈ y))
29	27, 28	orbi12d 706	. . . . . . . . 9 ⊢ (z = {∅} → ((x ∈ z ∨ z ∈ y) ↔ (x ∈ {∅} ∨ {∅} ∈ y)))
30	29	imbi2d 219	. . . . . . . 8 ⊢ (z = {∅} → ((x ∈ y → (x ∈ z ∨ z ∈ y)) ↔ (x ∈ y → (x ∈ {∅} ∨ {∅} ∈ y))))
31	26, 30	imbi12d 223	. . . . . . 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 4025	. . . . . . . . . . 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 133	. . . . . . . . . 10 ⊢ ( E Po On ∧ ∀x ∈ On ∀y ∈ On ∀z ∈ On (x E y → (x E z ∨ z E y)))
35	34	simpri 106	. . . . . . . . 9 ⊢ ∀x ∈ On ∀y ∈ On ∀z ∈ On (x E y → (x E z ∨ z E y))
36		epel 4020	. . . . . . . . . . . 12 ⊢ (x E y ↔ x ∈ y)
37		epel 4020	. . . . . . . . . . . . 13 ⊢ (x E z ↔ x ∈ z)
38		epel 4020	. . . . . . . . . . . . 13 ⊢ (z E y ↔ z ∈ y)
39	37, 38	orbi12i 680	. . . . . . . . . . . 12 ⊢ ((x E z ∨ z E y) ↔ (x ∈ z ∨ z ∈ y))
40	36, 39	imbi12i 228	. . . . . . . . . . 11 ⊢ ((x E y → (x E z ∨ z E y)) ↔ (x ∈ y → (x ∈ z ∨ z ∈ y)))
41	40	2ralbii 2326	. . . . . . . . . 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 2324	. . . . . . . . 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 133	. . . . . . . 8 ⊢ ∀x ∈ On ∀y ∈ On ∀z ∈ On (x ∈ y → (x ∈ z ∨ z ∈ y))
44	43	rspec3 2403	. . . . . . 7 ⊢ ((x ∈ On ∧ y ∈ On ∧ z ∈ On) → (x ∈ y → (x ∈ z ∨ z ∈ y)))
45	24, 31, 44	vtocl 2602	. . . . . 6 ⊢ ((x ∈ On ∧ y ∈ On ∧ {∅} ∈ On) → (x ∈ y → (x ∈ {∅} ∨ {∅} ∈ y)))
46	16, 23, 45	vtocl 2602	. . . . 5 ⊢ ((x ∈ On ∧ suc {w ∈ {∅} ∣ φ} ∈ On ∧ {∅} ∈ On) → (x ∈ suc {w ∈ {∅} ∣ φ} → (x ∈ {∅} ∨ {∅} ∈ suc {w ∈ {∅} ∣ φ})))
47	2, 15, 46	vtocl 2602	. . . 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 1231	. . 3 ⊢ ({w ∈ {∅} ∣ φ} ∈ suc {w ∈ {∅} ∣ φ} → ({w ∈ {∅} ∣ φ} ∈ {∅} ∨ {∅} ∈ suc {w ∈ {∅} ∣ φ}))
49	2	elsnc 3390	. . . . 5 ⊢ ({w ∈ {∅} ∣ φ} ∈ {∅} ↔ {w ∈ {∅} ∣ φ} = ∅)
50		ordtriexmidlem2 4209	. . . . 5 ⊢ ({w ∈ {∅} ∣ φ} = ∅ → ¬ φ)
51	49, 50	sylbi 114	. . . 4 ⊢ ({w ∈ {∅} ∣ φ} ∈ {∅} → ¬ φ)
52		elirr 4224	. . . . . . 7 ⊢ ¬ {∅} ∈ {∅}
53		elrabi 2689	. . . . . . 7 ⊢ ({∅} ∈ {w ∈ {∅} ∣ φ} → {∅} ∈ {∅})
54	52, 53	mto 587	. . . . . 6 ⊢ ¬ {∅} ∈ {w ∈ {∅} ∣ φ}
55		elsuci 4106	. . . . . . 7 ⊢ ({∅} ∈ suc {w ∈ {∅} ∣ φ} → ({∅} ∈ {w ∈ {∅} ∣ φ} ∨ {∅} = {w ∈ {∅} ∣ φ}))
56	55	ord 642	. . . . . 6 ⊢ ({∅} ∈ suc {w ∈ {∅} ∣ φ} → (¬ {∅} ∈ {w ∈ {∅} ∣ φ} → {∅} = {w ∈ {∅} ∣ φ}))
57	54, 56	mpi 15	. . . . 5 ⊢ ({∅} ∈ suc {w ∈ {∅} ∣ φ} → {∅} = {w ∈ {∅} ∣ φ})
58		0ex 3875	. . . . . . 7 ⊢ ∅ ∈ V
59		biidd 161	. . . . . . 7 ⊢ (w = ∅ → (φ ↔ φ))
60	58, 59	rabsnt 3436	. . . . . 6 ⊢ ({w ∈ {∅} ∣ φ} = {∅} → φ)
61	60	eqcoms 2040	. . . . 5 ⊢ ({∅} = {w ∈ {∅} ∣ φ} → φ)
62	57, 61	syl 14	. . . 4 ⊢ ({∅} ∈ suc {w ∈ {∅} ∣ φ} → φ)
63	51, 62	orim12i 675	. . 3 ⊢ (({w ∈ {∅} ∣ φ} ∈ {∅} ∨ {∅} ∈ suc {w ∈ {∅} ∣ φ}) → (¬ φ ∨ φ))
64	3, 48, 63	mp2b 8	. 2 ⊢ (¬ φ ∨ φ)
65		orcom 646	. 2 ⊢ ((¬ φ ∨ φ) ↔ (φ ∨ ¬ φ))
66	64, 65	mpbi 133	1 ⊢ (φ ∨ ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 628   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∀wral 2300  {crab 2304  ∅c0 3218  {csn 3367   class class class wbr 3755   E cep 4015   Po wpo 4022   Or wor 4023  Oncon0 4066  suc csuc 4068

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-tr 3846  df-eprel 4017  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074

This theorem is referenced by: (None)