Theorem exmidomni 7432

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 7431	. 2 ⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)
2		vex 2815	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
3		eleq1w 2293	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑥 ∈ Omni ↔ 𝑢 ∈ Omni))
4	2, 3	spcv 2910	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 ∈ Omni → 𝑢 ∈ Omni)
5		xpeq1 4762	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (𝑥 × {∅}) = (𝑢 × {∅}))
6	5	fveq1d 5671	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → ((𝑥 × {∅})‘𝑦) = ((𝑢 × {∅})‘𝑦))
7	6	eqeq1d 2241	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = ∅ ↔ ((𝑢 × {∅})‘𝑦) = ∅))
8	7	rexeqbi1dv 2753	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ↔ ∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅))
9	6	eqeq1d 2241	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = 1o ↔ ((𝑢 × {∅})‘𝑦) = 1o))
10	9	raleqbi1dv 2752	. . . . . . . . . . 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 2815	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
13		isomni 7426	. . . . . . . . . . . . 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 4236	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
17	16	prid1 3796	. . . . . . . . . . . . 13 ⊢ ∅ ∈ {∅, 1o}
18		df2o3 6661	. . . . . . . . . . . . 13 ⊢ 2o = {∅, 1o}
19	17, 18	eleqtrri 2308	. . . . . . . . . . . 12 ⊢ ∅ ∈ 2o
20	19	fconst6 5566	. . . . . . . . . . 11 ⊢ (𝑥 × {∅}):𝑥⟶2o
21		p0ex 4300	. . . . . . . . . . . . 13 ⊢ {∅} ∈ V
22	12, 21	xpex 4865	. . . . . . . . . . . 12 ⊢ (𝑥 × {∅}) ∈ V
23		feq1 5490	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑥 × {∅}) → (𝑓:𝑥⟶2o ↔ (𝑥 × {∅}):𝑥⟶2o))
24		fveq1 5668	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 × {∅}) → (𝑓‘𝑦) = ((𝑥 × {∅})‘𝑦))
25	24	eqeq1d 2241	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 × {∅}) → ((𝑓‘𝑦) = ∅ ↔ ((𝑥 × {∅})‘𝑦) = ∅))
26	25	rexbidv 2543	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 × {∅}) → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ↔ ∃𝑦 ∈ 𝑥 ((𝑥 × {∅})‘𝑦) = ∅))
27	24	eqeq1d 2241	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 × {∅}) → ((𝑓‘𝑦) = 1o ↔ ((𝑥 × {∅})‘𝑦) = 1o))
28	27	ralbidv 2542	. . . . . . . . . . . . . 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 2910	. . . . . . . . . . 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 2880	. . . . . . . . 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 3608	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → ∃𝑦 𝑦 ∈ 𝑢)
38		sssnm 3857	. . . . . . . . . . . 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 2573	. . . . . . . . . . . 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 2384	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝑢
47		nfcv 2384	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∅
48		1n0 6664	. . . . . . . . . . . . . 14 ⊢ 1o ≠ ∅
49	48	neii 2414	. . . . . . . . . . . . 13 ⊢ ¬ 1o = ∅
50		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
51	50	r19.21bi 2630	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦 ∈ 𝑢) → ((𝑢 × {∅})‘𝑦) = 1o)
52	16	fvconst2 5899	. . . . . . . . . . . . . . . 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 2267	. . . . . . . . . . . . . 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 3242	. . . . . . . . . 10 ⊢ (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦 ∈ 𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → 𝑢 ⊆ ∅)
59		ss0 3548	. . . . . . . . . 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 4310	. . 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 2203  ∀wral 2520  ∃wrex 2521  Vcvv 2812   ⊆ wss 3210  ∅c0 3507  {csn 3688  {cpr 3689  EXMIDwem 4306   × cxp 4746  ⟶wf 5347  ‘cfv 5351  1_oc1o 6639  2_oc2o 6640  Omnicomni 7424

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-exmid 4307  df-id 4413  df-suc 4491  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-1o 6646  df-2o 6647  df-omni 7425

This theorem is referenced by:  exmidlpo  7433