Theorem exmidomniim 7216

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 7217. (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 4230	. . . . . . . . 9 ⊢ (EXMID → DECID ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)
2		exmiddc 837	. . . . . . . . 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 730	. . . . . . 7 ⊢ (EXMID → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))
5	4	adantr 276	. . . . . 6 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2o) → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))
6		ffvelcdm 5698	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (𝑓‘𝑦) ∈ 2o)
7		df2o3 6497	. . . . . . . . . . . . . 14 ⊢ 2o = {∅, 1o}
8	6, 7	eleqtrdi 2289	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (𝑓‘𝑦) ∈ {∅, 1o})
9		elpri 3646	. . . . . . . . . . . . 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 725	. . . . . . . . . . 11 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (¬ (𝑓‘𝑦) = ∅ → (𝑓‘𝑦) = 1o))
12	11	ralimdva 2564	. . . . . . . . . 10 ⊢ (𝑓:𝑥⟶2o → (∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅ → ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))
13	12	con3d 632	. . . . . . . . 9 ⊢ (𝑓:𝑥⟶2o → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o → ¬ ∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅))
14	13	adantl 277	. . . . . . . 8 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2o) → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o → ¬ ∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅))
15		exmidexmid 4230	. . . . . . . . . 10 ⊢ (EXMID → DECID ∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅)
16		dfrex2dc 2488	. . . . . . . . . 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 276	. . . . . . . 8 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2o) → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ↔ ¬ ∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅))
19	14, 18	sylibrd 169	. . . . . . 7 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2o) → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o → ∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅))
20	19	orim1d 788	. . . . . 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 115	. . . 4 ⊢ (EXMID → (𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)))
23	22	alrimiv 1888	. . 3 ⊢ (EXMID → ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)))
24		isomni 7211	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))))
25	24	elv 2767	. . 3 ⊢ (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)))
26	23, 25	sylibr 134	. 2 ⊢ (EXMID → 𝑥 ∈ Omni)
27	26	alrimiv 1888	1 ⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835  ∀wal 1362   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  Vcvv 2763  ∅c0 3451  {cpr 3624  EXMIDwem 4228  ⟶wf 5255  ‘cfv 5259  1_oc1o 6476  2_oc2o 6477  Omnicomni 7209

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-exmid 4229  df-id 4329  df-suc 4407  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-1o 6483  df-2o 6484  df-omni 7210

This theorem is referenced by:  exmidomni  7217