Theorem exmidomni 7068

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 7067	. 2 ⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)
2		vex 2715	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
3		eleq1w 2218	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑥 ∈ Omni ↔ 𝑢 ∈ Omni))
4	2, 3	spcv 2806	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 ∈ Omni → 𝑢 ∈ Omni)
5		xpeq1 4597	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (𝑥 × {∅}) = (𝑢 × {∅}))
6	5	fveq1d 5467	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → ((𝑥 × {∅})‘𝑦) = ((𝑢 × {∅})‘𝑦))
7	6	eqeq1d 2166	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = ∅ ↔ ((𝑢 × {∅})‘𝑦) = ∅))
8	7	rexeqbi1dv 2661	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ↔ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅))
9	6	eqeq1d 2166	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = 1o ↔ ((𝑢 × {∅})‘𝑦) = 1o))
10	9	raleqbi1dv 2660	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o ↔ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
11	8, 10	orbi12d 783	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ((∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o) ↔ (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)))
12		vex 2715	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
13		isomni 7062	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))))
14	12, 13	ax-mp 5	. . . . . . . . . . . 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 4091	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
17	16	prid1 3665	. . . . . . . . . . . . 13 ⊢ ∅ ∈ {∅, 1o}
18		df2o3 6371	. . . . . . . . . . . . 13 ⊢ 2o = {∅, 1o}
19	17, 18	eleqtrri 2233	. . . . . . . . . . . 12 ⊢ ∅ ∈ 2o
20	19	fconst6 5366	. . . . . . . . . . 11 ⊢ (𝑥 × {∅}):𝑥⟶2o
21		p0ex 4148	. . . . . . . . . . . . 13 ⊢ {∅} ∈ V
22	12, 21	xpex 4698	. . . . . . . . . . . 12 ⊢ (𝑥 × {∅}) ∈ V
23		feq1 5299	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑥 × {∅}) → (𝑓:𝑥⟶2o ↔ (𝑥 × {∅}):𝑥⟶2o))
24		fveq1 5464	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 × {∅}) → (𝑓‘𝑦) = ((𝑥 × {∅})‘𝑦))
25	24	eqeq1d 2166	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 × {∅}) → ((𝑓‘𝑦) = ∅ ↔ ((𝑥 × {∅})‘𝑦) = ∅))
26	25	rexbidv 2458	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 × {∅}) → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ↔ ∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅))
27	24	eqeq1d 2166	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 × {∅}) → ((𝑓‘𝑦) = 1o ↔ ((𝑥 × {∅})‘𝑦) = 1o))
28	27	ralbidv 2457	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 × {∅}) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o ↔ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o))
29	26, 28	orbi12d 783	. . . . . . . . . . . . 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 2806	. . . . . . . . . . 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 1426	. . . . . . . . . 10 ⊢ (𝑥 ∈ Omni → (∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o))
33	11, 32	vtoclga 2778	. . . . . . . . 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 274	. . . . . . 7 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
36		simplr 520	. . . . . . . . . 10 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → 𝑢 ⊆ {∅})
37		rexm 3493	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → ∃𝑦 𝑦 ∈ 𝑢)
38		sssnm 3717	. . . . . . . . . . . 12 ⊢ (∃𝑦 𝑦 ∈ 𝑢 → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
39	37, 38	syl 14	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
40	39	adantl 275	. . . . . . . . . 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 1508	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅})
44		nfra1 2488	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o
45	43, 44	nfan 1545	. . . . . . . . . . 11 ⊢ Ⅎ𝑦((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
46		nfcv 2299	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝑢
47		nfcv 2299	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∅
48		1n0 6373	. . . . . . . . . . . . . 14 ⊢ 1o ≠ ∅
49	48	neii 2329	. . . . . . . . . . . . 13 ⊢ ¬ 1o = ∅
50		simpr 109	. . . . . . . . . . . . . . . 16 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
51	50	r19.21bi 2545	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦 ∈ 𝑢) → ((𝑢 × {∅})‘𝑦) = 1o)
52	16	fvconst2 5680	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑢 → ((𝑢 × {∅})‘𝑦) = ∅)
53	52	adantl 275	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦 ∈ 𝑢) → ((𝑢 × {∅})‘𝑦) = ∅)
54	51, 53	eqtr3d 2192	. . . . . . . . . . . . . 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 654	. . . . . . . . . . . 12 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ¬ 𝑦 ∈ 𝑢)
57	56	pm2.21d 609	. . . . . . . . . . 11 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → (𝑦 ∈ 𝑢 → 𝑦 ∈ ∅))
58	45, 46, 47, 57	ssrd 3133	. . . . . . . . . 10 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → 𝑢 ⊆ ∅)
59		ss0 3434	. . . . . . . . . 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 776	. . . . . . 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 719	. . . . 5 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (𝑢 = ∅ ∨ 𝑢 = {∅}))
65	64	ex 114	. . . 4 ⊢ (∀𝑥 𝑥 ∈ Omni → (𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
66	65	alrimiv 1854	. . 3 ⊢ (∀𝑥 𝑥 ∈ Omni → ∀𝑢(𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
67		exmid01 4158	. . 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 698  ∀wal 1333   = wceq 1335  ∃wex 1472   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436  Vcvv 2712   ⊆ wss 3102  ∅c0 3394  {csn 3560  {cpr 3561  EXMIDwem 4154   × cxp 4581  ⟶wf 5163  ‘cfv 5167  1_oc1o 6350  2_oc2o 6351  Omnicomni 7060

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-exmid 4155  df-id 4252  df-suc 4330  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-fv 5175  df-1o 6357  df-2o 6358  df-omni 7061

This theorem is referenced by:  exmidlpo  7069