Theorem exmidomniim 7250

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 7251. (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 4244	. . . . . . . . 9 ⊢ (EXMID → DECID ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)
2		exmiddc 838	. . . . . . . . 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 731	. . . . . . 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 5720	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (𝑓‘𝑦) ∈ 2o)
7		df2o3 6523	. . . . . . . . . . . . . 14 ⊢ 2o = {∅, 1o}
8	6, 7	eleqtrdi 2299	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (𝑓‘𝑦) ∈ {∅, 1o})
9		elpri 3657	. . . . . . . . . . . . 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 726	. . . . . . . . . . 11 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (¬ (𝑓‘𝑦) = ∅ → (𝑓‘𝑦) = 1o))
12	11	ralimdva 2574	. . . . . . . . . 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 4244	. . . . . . . . . 10 ⊢ (EXMID → DECID ∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅)
16		dfrex2dc 2498	. . . . . . . . . 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 789	. . . . . 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 1898	. . 3 ⊢ (EXMID → ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)))
24		isomni 7245	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))))
25	24	elv 2777	. . 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 1898	1 ⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710  DECID wdc 836  ∀wal 1371   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  Vcvv 2773  ∅c0 3461  {cpr 3635  EXMIDwem 4242  ⟶wf 5272  ‘cfv 5276  1_oc1o 6502  2_oc2o 6503  Omnicomni 7243

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3000  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-exmid 4243  df-id 4344  df-suc 4422  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-fv 5284  df-1o 6509  df-2o 6510  df-omni 7244

This theorem is referenced by:  exmidomni  7251