Theorem exmidomni 6852

Description: Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.)

Assertion

Ref	Expression
exmidomni	⊢ (EXMID ↔ ∀𝑥 𝑥 ∈ Omni)

Proof of Theorem exmidomni

Dummy variables 𝑢 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidomniim 6851	. 2 ⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)
2		vex 2623	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
3		eleq1w 2149	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑥 ∈ Omni ↔ 𝑢 ∈ Omni))
4	2, 3	spcv 2713	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 ∈ Omni → 𝑢 ∈ Omni)
5		xpeq1 4465	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (𝑥 × {∅}) = (𝑢 × {∅}))
6	5	fveq1d 5320	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → ((𝑥 × {∅})‘𝑦) = ((𝑢 × {∅})‘𝑦))
7	6	eqeq1d 2097	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = ∅ ↔ ((𝑢 × {∅})‘𝑦) = ∅))
8	7	rexeqbi1dv 2572	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ↔ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅))
9	6	eqeq1d 2097	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = 1o ↔ ((𝑢 × {∅})‘𝑦) = 1o))
10	9	raleqbi1dv 2571	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o ↔ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
11	8, 10	orbi12d 743	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ((∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o) ↔ (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)))
12		vex 2623	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
13		isomni 6846	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))))
14	12, 13	ax-mp 7	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)))
15	14	biimpi 119	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Omni → ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)))
16		0ex 3972	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
17	16	prid1 3552	. . . . . . . . . . . . 13 ⊢ ∅ ∈ {∅, 1o}
18		df2o3 6209	. . . . . . . . . . . . 13 ⊢ 2o = {∅, 1o}
19	17, 18	eleqtrri 2164	. . . . . . . . . . . 12 ⊢ ∅ ∈ 2o
20	19	fconst6 5223	. . . . . . . . . . 11 ⊢ (𝑥 × {∅}):𝑥⟶2o
21		p0ex 4029	. . . . . . . . . . . . 13 ⊢ {∅} ∈ V
22	12, 21	xpex 4566	. . . . . . . . . . . 12 ⊢ (𝑥 × {∅}) ∈ V
23		feq1 5158	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑥 × {∅}) → (𝑓:𝑥⟶2o ↔ (𝑥 × {∅}):𝑥⟶2o))
24		fveq1 5317	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 × {∅}) → (𝑓‘𝑦) = ((𝑥 × {∅})‘𝑦))
25	24	eqeq1d 2097	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 × {∅}) → ((𝑓‘𝑦) = ∅ ↔ ((𝑥 × {∅})‘𝑦) = ∅))
26	25	rexbidv 2382	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 × {∅}) → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ↔ ∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅))
27	24	eqeq1d 2097	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 × {∅}) → ((𝑓‘𝑦) = 1o ↔ ((𝑥 × {∅})‘𝑦) = 1o))
28	27	ralbidv 2381	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 × {∅}) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o ↔ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o))
29	26, 28	orbi12d 743	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑥 × {∅}) → ((∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o) ↔ (∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o)))
30	23, 29	imbi12d 233	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑥 × {∅}) → ((𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)) ↔ ((𝑥 × {∅}):𝑥⟶2o → (∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o))))
31	22, 30	spcv 2713	. . . . . . . . . . 11 ⊢ (∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)) → ((𝑥 × {∅}):𝑥⟶2o → (∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o)))
32	15, 20, 31	mpisyl 1381	. . . . . . . . . 10 ⊢ (𝑥 ∈ Omni → (∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o))
33	11, 32	vtoclga 2686	. . . . . . . . 9 ⊢ (𝑢 ∈ Omni → (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
34	4, 33	syl 14	. . . . . . . 8 ⊢ (∀𝑥 𝑥 ∈ Omni → (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
35	34	adantr 271	. . . . . . 7 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
36		simplr 498	. . . . . . . . . 10 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → 𝑢 ⊆ {∅})
37		rexm 3385	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → ∃𝑦 𝑦 ∈ 𝑢)
38		sssnm 3604	. . . . . . . . . . . 12 ⊢ (∃𝑦 𝑦 ∈ 𝑢 → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
39	37, 38	syl 14	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
40	39	adantl 272	. . . . . . . . . 10 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
41	36, 40	mpbid 146	. . . . . . . . 9 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → 𝑢 = {∅})
42	41	ex 114	. . . . . . . 8 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → 𝑢 = {∅}))
43		nfv 1467	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅})
44		nfra1 2410	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o
45	43, 44	nfan 1503	. . . . . . . . . . 11 ⊢ Ⅎ𝑦((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
46		nfcv 2229	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝑢
47		nfcv 2229	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∅
48		1n0 6211	. . . . . . . . . . . . . 14 ⊢ 1o ≠ ∅
49	48	neii 2258	. . . . . . . . . . . . 13 ⊢ ¬ 1o = ∅
50		simpr 109	. . . . . . . . . . . . . . . 16 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
51	50	r19.21bi 2462	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦 ∈ 𝑢) → ((𝑢 × {∅})‘𝑦) = 1o)
52	16	fvconst2 5527	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑢 → ((𝑢 × {∅})‘𝑦) = ∅)
53	52	adantl 272	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦 ∈ 𝑢) → ((𝑢 × {∅})‘𝑦) = ∅)
54	51, 53	eqtr3d 2123	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦 ∈ 𝑢) → 1o = ∅)
55	54	ex 114	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → (𝑦 ∈ 𝑢 → 1o = ∅))
56	49, 55	mtoi 626	. . . . . . . . . . . 12 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ¬ 𝑦 ∈ 𝑢)
57	56	pm2.21d 585	. . . . . . . . . . 11 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → (𝑦 ∈ 𝑢 → 𝑦 ∈ ∅))
58	45, 46, 47, 57	ssrd 3031	. . . . . . . . . 10 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → 𝑢 ⊆ ∅)
59		ss0 3327	. . . . . . . . . 10 ⊢ (𝑢 ⊆ ∅ → 𝑢 = ∅)
60	58, 59	syl 14	. . . . . . . . 9 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → 𝑢 = ∅)
61	60	ex 114	. . . . . . . 8 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o → 𝑢 = ∅))
62	42, 61	orim12d 736	. . . . . . 7 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → ((∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → (𝑢 = {∅} ∨ 𝑢 = ∅)))
63	35, 62	mpd 13	. . . . . 6 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (𝑢 = {∅} ∨ 𝑢 = ∅))
64	63	orcomd 684	. . . . 5 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (𝑢 = ∅ ∨ 𝑢 = {∅}))
65	64	ex 114	. . . 4 ⊢ (∀𝑥 𝑥 ∈ Omni → (𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
66	65	alrimiv 1803	. . 3 ⊢ (∀𝑥 𝑥 ∈ Omni → ∀𝑢(𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
67		exmid01 4038	. . 3 ⊢ (EXMID ↔ ∀𝑢(𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
68	66, 67	sylibr 133	. 2 ⊢ (∀𝑥 𝑥 ∈ Omni → EXMID)
69	1, 68	impbii 125	1 ⊢ (EXMID ↔ ∀𝑥 𝑥 ∈ Omni)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 665  ∀wal 1288   = wceq 1290  ∃wex 1427   ∈ wcel 1439  ∀wral 2360  ∃wrex 2361  Vcvv 2620   ⊆ wss 3000  ∅c0 3287  {csn 3450  {cpr 3451  EXMIDwem 4035   × cxp 4449  ⟶wf 5024  ‘cfv 5028  1_oc1o 6188  2_oc2o 6189  Omnicomni 6842

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-exmid 4036  df-id 4129  df-suc 4207  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-1o 6195  df-2o 6196  df-omni 6844

This theorem is referenced by: (None)