Theorem exmidomniim 7340

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 7341. (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 4286	. . . . . . . . 9 ⊢ (EXMID → DECID ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)
2		exmiddc 843	. . . . . . . . 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 736	. . . . . . 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 5780	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (𝑓‘𝑦) ∈ 2o)
7		df2o3 6597	. . . . . . . . . . . . . 14 ⊢ 2o = {∅, 1o}
8	6, 7	eleqtrdi 2324	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (𝑓‘𝑦) ∈ {∅, 1o})
9		elpri 3692	. . . . . . . . . . . . 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 731	. . . . . . . . . . 11 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (¬ (𝑓‘𝑦) = ∅ → (𝑓‘𝑦) = 1o))
12	11	ralimdva 2599	. . . . . . . . . 10 ⊢ (𝑓:𝑥⟶2o → (∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅ → ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))
13	12	con3d 636	. . . . . . . . 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 4286	. . . . . . . . . 10 ⊢ (EXMID → DECID ∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅)
16		dfrex2dc 2523	. . . . . . . . . 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 794	. . . . . 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 1922	. . 3 ⊢ (EXMID → ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)))
24		isomni 7335	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))))
25	24	elv 2806	. . 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 1922	1 ⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715  DECID wdc 841  ∀wal 1395   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  Vcvv 2802  ∅c0 3494  {cpr 3670  EXMIDwem 4284  ⟶wf 5322  ‘cfv 5326  1_oc1o 6575  2_oc2o 6576  Omnicomni 7333

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-exmid 4285  df-id 4390  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-1o 6582  df-2o 6583  df-omni 7334

This theorem is referenced by:  exmidomni  7341