Theorem exmidomniim 6795

Description: Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 6796. (Contributed by Jim Kingdon, 29-Jun-2022.)

Assertion

Ref	Expression
exmidomniim	⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)

Proof of Theorem exmidomniim

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

Step	Hyp	Ref	Expression
1		exmidexmid 4031	. . . . . . . . 9 ⊢ (EXMID → DECID ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜)
2		exmiddc 782	. . . . . . . . 9 ⊢ (DECID ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜 ∨ ¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜))
3	1, 2	syl 14	. . . . . . . 8 ⊢ (EXMID → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜 ∨ ¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜))
4	3	orcomd 683	. . . . . . 7 ⊢ (EXMID → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜 ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜))
5	4	adantr 270	. . . . . 6 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2𝑜) → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜 ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜))
6		ffvelrn 5432	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑥⟶2𝑜 ∧ 𝑦 ∈ 𝑥) → (𝑓‘𝑦) ∈ 2𝑜)
7		df2o3 6195	. . . . . . . . . . . . . 14 ⊢ 2𝑜 = {∅, 1𝑜}
8	6, 7	syl6eleq 2180	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑥⟶2𝑜 ∧ 𝑦 ∈ 𝑥) → (𝑓‘𝑦) ∈ {∅, 1𝑜})
9		elpri 3469	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑦) ∈ {∅, 1𝑜} → ((𝑓‘𝑦) = ∅ ∨ (𝑓‘𝑦) = 1𝑜))
10	8, 9	syl 14	. . . . . . . . . . . 12 ⊢ ((𝑓:𝑥⟶2𝑜 ∧ 𝑦 ∈ 𝑥) → ((𝑓‘𝑦) = ∅ ∨ (𝑓‘𝑦) = 1𝑜))
11	10	ord 678	. . . . . . . . . . 11 ⊢ ((𝑓:𝑥⟶2𝑜 ∧ 𝑦 ∈ 𝑥) → (¬ (𝑓‘𝑦) = ∅ → (𝑓‘𝑦) = 1𝑜))
12	11	ralimdva 2441	. . . . . . . . . 10 ⊢ (𝑓:𝑥⟶2𝑜 → (∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅ → ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜))
13	12	con3d 596	. . . . . . . . 9 ⊢ (𝑓:𝑥⟶2𝑜 → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜 → ¬ ∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅))
14	13	adantl 271	. . . . . . . 8 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2𝑜) → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜 → ¬ ∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅))
15		exmidexmid 4031	. . . . . . . . . 10 ⊢ (EXMID → DECID ∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅)
16		dfrex2dc 2371	. . . . . . . . . 10 ⊢ (DECID ∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ↔ ¬ ∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅))
17	15, 16	syl 14	. . . . . . . . 9 ⊢ (EXMID → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ↔ ¬ ∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅))
18	17	adantr 270	. . . . . . . 8 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2𝑜) → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ↔ ¬ ∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅))
19	14, 18	sylibrd 167	. . . . . . 7 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2𝑜) → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜 → ∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅))
20	19	orim1d 736	. . . . . 6 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2𝑜) → ((¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜 ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜) → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜)))
21	5, 20	mpd 13	. . . . 5 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2𝑜) → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜))
22	21	ex 113	. . . 4 ⊢ (EXMID → (𝑓:𝑥⟶2𝑜 → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜)))
23	22	alrimiv 1802	. . 3 ⊢ (EXMID → ∀𝑓(𝑓:𝑥⟶2𝑜 → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜)))
24		vex 2622	. . . 4 ⊢ 𝑥 ∈ V
25		isomni 6790	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2𝑜 → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜))))
26	24, 25	ax-mp 7	. . 3 ⊢ (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2𝑜 → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1𝑜)))
27	23, 26	sylibr 132	. 2 ⊢ (EXMID → 𝑥 ∈ Omni)
28	27	alrimiv 1802	1 ⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 664  DECID wdc 780  ∀wal 1287   = wceq 1289   ∈ wcel 1438  ∀wral 2359  ∃wrex 2360  Vcvv 2619  ∅c0 3286  {cpr 3447  EXMIDwem 4029  ⟶wf 5011  ‘cfv 5015  1_𝑜c1o 6174  2_𝑜c2o 6175  Omnicomni 6786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-exmid 4030  df-id 4120  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-1o 6181  df-2o 6182  df-omni 6788

This theorem is referenced by:  exmidomni  6796