Theorem ordsoexmid 4610

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 4567	. . . . 5 ⊢ {𝑤 ∈ {∅} ∣ 𝜑} ∈ On
2	1	elexi 2784	. . . 4 ⊢ {𝑤 ∈ {∅} ∣ 𝜑} ∈ V
3	2	sucid 4464	. . 3 ⊢ {𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}
4	1	onsuci 4564	. . . 4 ⊢ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On
5		suc0 4458	. . . . 5 ⊢ suc ∅ = {∅}
6		0elon 4439	. . . . . 6 ⊢ ∅ ∈ On
7	6	onsuci 4564	. . . . 5 ⊢ suc ∅ ∈ On
8	5, 7	eqeltrri 2279	. . . 4 ⊢ {∅} ∈ On
9		eleq1 2268	. . . . . . 7 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ On ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ On))
10	9	3anbi1d 1329	. . . . . 6 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On)))
11		eleq1 2268	. . . . . . 7 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
12		eleq1 2268	. . . . . . . 8 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅}))
13	12	orbi1d 793	. . . . . . 7 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
14	11, 13	imbi12d 234	. . . . . 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 234	. . . . 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 2784	. . . . . 6 ⊢ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ V
17		eleq1 2268	. . . . . . . 8 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑦 ∈ On ↔ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On))
18	17	3anbi2d 1330	. . . . . . 7 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On) ↔ (𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On)))
19		eleq2 2269	. . . . . . . 8 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
20		eleq2 2269	. . . . . . . . 9 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ({∅} ∈ 𝑦 ↔ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
21	20	orbi2d 792	. . . . . . . 8 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦) ↔ (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
22	19, 21	imbi12d 234	. . . . . . 7 ⊢ (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ 𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)) ↔ (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))))
23	18, 22	imbi12d 234	. . . . . 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 4232	. . . . . . 7 ⊢ {∅} ∈ V
25		eleq1 2268	. . . . . . . . 9 ⊢ (𝑧 = {∅} → (𝑧 ∈ On ↔ {∅} ∈ On))
26	25	3anbi3d 1331	. . . . . . . 8 ⊢ (𝑧 = {∅} → ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑧 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On)))
27		eleq2 2269	. . . . . . . . . 10 ⊢ (𝑧 = {∅} → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ {∅}))
28		eleq1 2268	. . . . . . . . . 10 ⊢ (𝑧 = {∅} → (𝑧 ∈ 𝑦 ↔ {∅} ∈ 𝑦))
29	27, 28	orbi12d 795	. . . . . . . . 9 ⊢ (𝑧 = {∅} → ((𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦) ↔ (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)))
30	29	imbi2d 230	. . . . . . . 8 ⊢ (𝑧 = {∅} → ((𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)) ↔ (𝑥 ∈ 𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦))))
31	26, 30	imbi12d 234	. . . . . . 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 4344	. . . . . . . . . . 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 145	. . . . . . . . . 10 ⊢ ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧 ∨ 𝑧 E 𝑦)))
35	34	simpri 113	. . . . . . . . 9 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧 ∨ 𝑧 E 𝑦))
36		epel 4339	. . . . . . . . . . . 12 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
37		epel 4339	. . . . . . . . . . . . 13 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧)
38		epel 4339	. . . . . . . . . . . . 13 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
39	37, 38	orbi12i 766	. . . . . . . . . . . 12 ⊢ ((𝑥 E 𝑧 ∨ 𝑧 E 𝑦) ↔ (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))
40	36, 39	imbi12i 239	. . . . . . . . . . 11 ⊢ ((𝑥 E 𝑦 → (𝑥 E 𝑧 ∨ 𝑧 E 𝑦)) ↔ (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)))
41	40	2ralbii 2514	. . . . . . . . . 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 2512	. . . . . . . . 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 145	. . . . . . . 8 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))
44	43	rspec3 2596	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)))
45	24, 31, 44	vtocl 2827	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On) → (𝑥 ∈ 𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)))
46	16, 23, 45	vtocl 2827	. . . . 5 ⊢ ((𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
47	2, 15, 46	vtocl 2827	. . . 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 1350	. . 3 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
49	2	elsn 3649	. . . . 5 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ↔ {𝑤 ∈ {∅} ∣ 𝜑} = ∅)
50		ordtriexmidlem2 4568	. . . . 5 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
51	49, 50	sylbi 121	. . . 4 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
52		elirr 4589	. . . . . . 7 ⊢ ¬ {∅} ∈ {∅}
53		elrabi 2926	. . . . . . 7 ⊢ ({∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑} → {∅} ∈ {∅})
54	52, 53	mto 664	. . . . . 6 ⊢ ¬ {∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑}
55		elsuci 4450	. . . . . . 7 ⊢ ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑤 ∈ {∅} ∣ 𝜑}))
56	55	ord 726	. . . . . 6 ⊢ ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (¬ {∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑} → {∅} = {𝑤 ∈ {∅} ∣ 𝜑}))
57	54, 56	mpi 15	. . . . 5 ⊢ ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → {∅} = {𝑤 ∈ {∅} ∣ 𝜑})
58		0ex 4171	. . . . . . 7 ⊢ ∅ ∈ V
59		biidd 172	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝜑 ↔ 𝜑))
60	58, 59	rabsnt 3708	. . . . . 6 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} = {∅} → 𝜑)
61	60	eqcoms 2208	. . . . 5 ⊢ ({∅} = {𝑤 ∈ {∅} ∣ 𝜑} → 𝜑)
62	57, 61	syl 14	. . . 4 ⊢ ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → 𝜑)
63	51, 62	orim12i 761	. . 3 ⊢ (({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑))
64	3, 48, 63	mp2b 8	. 2 ⊢ (¬ 𝜑 ∨ 𝜑)
65		orcom 730	. 2 ⊢ ((¬ 𝜑 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
66	64, 65	mpbi 145	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 710   ∧ w3a 981   = wceq 1373   ∈ wcel 2176  ∀wral 2484  {crab 2488  ∅c0 3460  {csn 3633   class class class wbr 4044   E cep 4334   Po wpo 4341   Or wor 4342  Oncon0 4410  suc csuc 4412

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-tr 4143  df-eprel 4336  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418

This theorem is referenced by: (None)