Theorem exmidomniim 7021

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 7022. (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 4128	. . . . . . . . 9 ⊢ (EXMID → DECID ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)
2		exmiddc 822	. . . . . . . . 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 719	. . . . . . 7 ⊢ (EXMID → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))
5	4	adantr 274	. . . . . 6 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2o) → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))
6		ffvelrn 5561	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (𝑓‘𝑦) ∈ 2o)
7		df2o3 6335	. . . . . . . . . . . . . 14 ⊢ 2o = {∅, 1o}
8	6, 7	eleqtrdi 2233	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (𝑓‘𝑦) ∈ {∅, 1o})
9		elpri 3555	. . . . . . . . . . . . 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 714	. . . . . . . . . . 11 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (¬ (𝑓‘𝑦) = ∅ → (𝑓‘𝑦) = 1o))
12	11	ralimdva 2502	. . . . . . . . . 10 ⊢ (𝑓:𝑥⟶2o → (∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅ → ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))
13	12	con3d 621	. . . . . . . . 9 ⊢ (𝑓:𝑥⟶2o → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o → ¬ ∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅))
14	13	adantl 275	. . . . . . . 8 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2o) → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o → ¬ ∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅))
15		exmidexmid 4128	. . . . . . . . . 10 ⊢ (EXMID → DECID ∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅)
16		dfrex2dc 2429	. . . . . . . . . 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 274	. . . . . . . 8 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2o) → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ↔ ¬ ∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅))
19	14, 18	sylibrd 168	. . . . . . 7 ⊢ ((EXMID ∧ 𝑓:𝑥⟶2o) → (¬ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o → ∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅))
20	19	orim1d 777	. . . . . 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 114	. . . 4 ⊢ (EXMID → (𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)))
23	22	alrimiv 1847	. . 3 ⊢ (EXMID → ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)))
24		isomni 7016	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))))
25	24	elv 2693	. . 3 ⊢ (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)))
26	23, 25	sylibr 133	. 2 ⊢ (EXMID → 𝑥 ∈ Omni)
27	26	alrimiv 1847	1 ⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 820  ∀wal 1330   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  Vcvv 2689  ∅c0 3368  {cpr 3533  EXMIDwem 4126  ⟶wf 5127  ‘cfv 5131  1_oc1o 6314  2_oc2o 6315  Omnicomni 7012

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-exmid 4127  df-id 4223  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-1o 6321  df-2o 6322  df-omni 7014

This theorem is referenced by:  exmidomni  7022