Theorem exmidomni 7118

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 7117	. 2 ⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)
2		vex 2733	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
3		eleq1w 2231	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑥 ∈ Omni ↔ 𝑢 ∈ Omni))
4	2, 3	spcv 2824	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 ∈ Omni → 𝑢 ∈ Omni)
5		xpeq1 4625	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (𝑥 × {∅}) = (𝑢 × {∅}))
6	5	fveq1d 5498	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → ((𝑥 × {∅})‘𝑦) = ((𝑢 × {∅})‘𝑦))
7	6	eqeq1d 2179	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = ∅ ↔ ((𝑢 × {∅})‘𝑦) = ∅))
8	7	rexeqbi1dv 2674	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ↔ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅))
9	6	eqeq1d 2179	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = 1o ↔ ((𝑢 × {∅})‘𝑦) = 1o))
10	9	raleqbi1dv 2673	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o ↔ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
11	8, 10	orbi12d 788	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ((∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o) ↔ (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)))
12		vex 2733	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
13		isomni 7112	. . . . . . . . . . . . 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 4116	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
17	16	prid1 3689	. . . . . . . . . . . . 13 ⊢ ∅ ∈ {∅, 1o}
18		df2o3 6409	. . . . . . . . . . . . 13 ⊢ 2o = {∅, 1o}
19	17, 18	eleqtrri 2246	. . . . . . . . . . . 12 ⊢ ∅ ∈ 2o
20	19	fconst6 5397	. . . . . . . . . . 11 ⊢ (𝑥 × {∅}):𝑥⟶2o
21		p0ex 4174	. . . . . . . . . . . . 13 ⊢ {∅} ∈ V
22	12, 21	xpex 4726	. . . . . . . . . . . 12 ⊢ (𝑥 × {∅}) ∈ V
23		feq1 5330	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑥 × {∅}) → (𝑓:𝑥⟶2o ↔ (𝑥 × {∅}):𝑥⟶2o))
24		fveq1 5495	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 × {∅}) → (𝑓‘𝑦) = ((𝑥 × {∅})‘𝑦))
25	24	eqeq1d 2179	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 × {∅}) → ((𝑓‘𝑦) = ∅ ↔ ((𝑥 × {∅})‘𝑦) = ∅))
26	25	rexbidv 2471	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 × {∅}) → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ↔ ∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅))
27	24	eqeq1d 2179	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 × {∅}) → ((𝑓‘𝑦) = 1o ↔ ((𝑥 × {∅})‘𝑦) = 1o))
28	27	ralbidv 2470	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 × {∅}) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o ↔ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o))
29	26, 28	orbi12d 788	. . . . . . . . . . . . 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 2824	. . . . . . . . . . 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 1439	. . . . . . . . . 10 ⊢ (𝑥 ∈ Omni → (∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = 1o))
33	11, 32	vtoclga 2796	. . . . . . . . 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 525	. . . . . . . . . 10 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → 𝑢 ⊆ {∅})
37		rexm 3514	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → ∃𝑦 𝑦 ∈ 𝑢)
38		sssnm 3741	. . . . . . . . . . . 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 1521	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅})
44		nfra1 2501	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o
45	43, 44	nfan 1558	. . . . . . . . . . 11 ⊢ Ⅎ𝑦((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
46		nfcv 2312	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝑢
47		nfcv 2312	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∅
48		1n0 6411	. . . . . . . . . . . . . 14 ⊢ 1o ≠ ∅
49	48	neii 2342	. . . . . . . . . . . . 13 ⊢ ¬ 1o = ∅
50		simpr 109	. . . . . . . . . . . . . . . 16 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
51	50	r19.21bi 2558	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦 ∈ 𝑢) → ((𝑢 × {∅})‘𝑦) = 1o)
52	16	fvconst2 5712	. . . . . . . . . . . . . . . 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 2205	. . . . . . . . . . . . . 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 659	. . . . . . . . . . . 12 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ¬ 𝑦 ∈ 𝑢)
57	56	pm2.21d 614	. . . . . . . . . . 11 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → (𝑦 ∈ 𝑢 → 𝑦 ∈ ∅))
58	45, 46, 47, 57	ssrd 3152	. . . . . . . . . 10 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → 𝑢 ⊆ ∅)
59		ss0 3455	. . . . . . . . . 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 781	. . . . . . 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 724	. . . . 5 ⊢ ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (𝑢 = ∅ ∨ 𝑢 = {∅}))
65	64	ex 114	. . . 4 ⊢ (∀𝑥 𝑥 ∈ Omni → (𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
66	65	alrimiv 1867	. . 3 ⊢ (∀𝑥 𝑥 ∈ Omni → ∀𝑢(𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
67		exmid01 4184	. . 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 703  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  Vcvv 2730   ⊆ wss 3121  ∅c0 3414  {csn 3583  {cpr 3584  EXMIDwem 4180   × cxp 4609  ⟶wf 5194  ‘cfv 5198  1_oc1o 6388  2_oc2o 6389  Omnicomni 7110

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-exmid 4181  df-id 4278  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-1o 6395  df-2o 6396  df-omni 7111

This theorem is referenced by:  exmidlpo  7119