Theorem exmidomni 7435

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 7434	. 2 ⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)
2		vex 2818	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
3		eleq1w 2295	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑥 ∈ Omni ↔ 𝑢 ∈ Omni))
4	2, 3	spcv 2913	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 ∈ Omni → 𝑢 ∈ Omni)
5		xpeq1 4765	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (𝑥 × {∅}) = (𝑢 × {∅}))
6	5	fveq1d 5674	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → ((𝑥 × {∅})‘𝑦) = ((𝑢 × {∅})‘𝑦))
7	6	eqeq1d 2243	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = ∅ ↔ ((𝑢 × {∅})‘𝑦) = ∅))
8	7	rexeqbi1dv 2756	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ↔ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅))
9	6	eqeq1d 2243	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = 1o ↔ ((𝑢 × {∅})‘𝑦) = 1o))
10	9	raleqbi1dv 2755	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o ↔ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
11	8, 10	orbi12d 801	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ((∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o) ↔ (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)))
12		vex 2818	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
13		isomni 7429	. . . . . . . . . . . . 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 120	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Omni → ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)))
16		0ex 4239	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
17	16	prid1 3799	. . . . . . . . . . . . 13 ⊢ ∅ ∈ {∅, 1o}
18		df2o3 6664	. . . . . . . . . . . . 13 ⊢ 2o = {∅, 1o}
19	17, 18	eleqtrri 2310	. . . . . . . . . . . 12 ⊢ ∅ ∈ 2o
20	19	fconst6 5569	. . . . . . . . . . 11 ⊢ (𝑥 × {∅}):𝑥⟶2o
21		p0ex 4303	. . . . . . . . . . . . 13 ⊢ {∅} ∈ V
22	12, 21	xpex 4868	. . . . . . . . . . . 12 ⊢ (𝑥 × {∅}) ∈ V
23		feq1 5493	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑥 × {∅}) → (𝑓:𝑥⟶2o ↔ (𝑥 × {∅}):𝑥⟶2o))
24		fveq1 5671	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 × {∅}) → (𝑓‘𝑦) = ((𝑥 × {∅})‘𝑦))
25	24	eqeq1d 2243	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 × {∅}) → ((𝑓‘𝑦) = ∅ ↔ ((𝑥 × {∅})‘𝑦) = ∅))
26	25	rexbidv 2545	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 × {∅}) → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ↔ ∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅))
27	24	eqeq1d 2243	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 × {∅}) → ((𝑓‘𝑦) = 1o ↔ ((𝑥 × {∅})‘𝑦) = 1o))
28	27	ralbidv 2544	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 × {∅}) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o ↔ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o))
29	26, 28	orbi12d 801	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 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 2913	. . . . . . . . . . 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 1492	. . . . . . . . . 10 ⊢ (𝑥 ∈ Omni → (∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o))
33	11, 32	vtoclga 2883	. . . . . . . . 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 276	. . . . . . 7 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
36		simplr 529	. . . . . . . . . 10 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → 𝑢 ⊆ {∅})
37		rexm 3611	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → ∃𝑦 𝑦 ∈ 𝑢)
38		sssnm 3860	. . . . . . . . . . . 12 ⊢ (∃𝑦 𝑦 ∈ 𝑢 → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
39	37, 38	syl 14	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
40	39	adantl 277	. . . . . . . . . 10 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
41	36, 40	mpbid 147	. . . . . . . . 9 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → 𝑢 = {∅})
42	41	ex 115	. . . . . . . 8 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → 𝑢 = {∅}))
43		nfv 1577	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅})
44		nfra1 2575	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o
45	43, 44	nfan 1614	. . . . . . . . . . 11 ⊢ Ⅎ𝑦((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
46		nfcv 2386	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝑢
47		nfcv 2386	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∅
48		1n0 6667	. . . . . . . . . . . . . 14 ⊢ 1o ≠ ∅
49	48	neii 2416	. . . . . . . . . . . . 13 ⊢ ¬ 1o = ∅
50		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
51	50	r19.21bi 2632	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦 ∈ 𝑢) → ((𝑢 × {∅})‘𝑦) = 1o)
52	16	fvconst2 5902	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑢 → ((𝑢 × {∅})‘𝑦) = ∅)
53	52	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦 ∈ 𝑢) → ((𝑢 × {∅})‘𝑦) = ∅)
54	51, 53	eqtr3d 2269	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦 ∈ 𝑢) → 1o = ∅)
55	54	ex 115	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → (𝑦 ∈ 𝑢 → 1o = ∅))
56	49, 55	mtoi 670	. . . . . . . . . . . 12 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ¬ 𝑦 ∈ 𝑢)
57	56	pm2.21d 624	. . . . . . . . . . 11 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → (𝑦 ∈ 𝑢 → 𝑦 ∈ ∅))
58	45, 46, 47, 57	ssrd 3245	. . . . . . . . . 10 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → 𝑢 ⊆ ∅)
59		ss0 3551	. . . . . . . . . 10 ⊢ (𝑢 ⊆ ∅ → 𝑢 = ∅)
60	58, 59	syl 14	. . . . . . . . 9 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → 𝑢 = ∅)
61	60	ex 115	. . . . . . . 8 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o → 𝑢 = ∅))
62	42, 61	orim12d 794	. . . . . . 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 737	. . . . 5 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (𝑢 = ∅ ∨ 𝑢 = {∅}))
65	64	ex 115	. . . 4 ⊢ (∀𝑥 𝑥 ∈ Omni → (𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
66	65	alrimiv 1923	. . 3 ⊢ (∀𝑥 𝑥 ∈ Omni → ∀𝑢(𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
67		exmid01 4313	. . 3 ⊢ (EXMID ↔ ∀𝑢(𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
68	66, 67	sylibr 134	. 2 ⊢ (∀𝑥 𝑥 ∈ Omni → EXMID)
69	1, 68	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥 𝑥 ∈ Omni)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  Vcvv 2815   ⊆ wss 3213  ∅c0 3510  {csn 3691  {cpr 3692  EXMIDwem 4309   × cxp 4749  ⟶wf 5350  ‘cfv 5354  1_oc1o 6642  2_oc2o 6643  Omnicomni 7427

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-exmid 4310  df-id 4416  df-suc 4494  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-1o 6649  df-2o 6650  df-omni 7428

This theorem is referenced by:  exmidlpo  7436