Theorem exmidomniim 7133

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 7134. (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 4193	. . . . . . . . 9 ⊢ (EXMID → DECID ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)
2		exmiddc 836	. . . . . . . . 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 729	. . . . . . 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 5645	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (𝑓‘𝑦) ∈ 2o)
7		df2o3 6425	. . . . . . . . . . . . . 14 ⊢ 2o = {∅, 1o}
8	6, 7	eleqtrdi 2270	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (𝑓‘𝑦) ∈ {∅, 1o})
9		elpri 3614	. . . . . . . . . . . . 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 724	. . . . . . . . . . 11 ⊢ ((𝑓:𝑥⟶2o ∧ 𝑦 ∈ 𝑥) → (¬ (𝑓‘𝑦) = ∅ → (𝑓‘𝑦) = 1o))
12	11	ralimdva 2544	. . . . . . . . . 10 ⊢ (𝑓:𝑥⟶2o → (∀𝑦 ∈ 𝑥 ¬ (𝑓‘𝑦) = ∅ → ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))
13	12	con3d 631	. . . . . . . . 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 4193	. . . . . . . . . 10 ⊢ (EXMID → DECID ∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅)
16		dfrex2dc 2468	. . . . . . . . . 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 787	. . . . . 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 1874	. . 3 ⊢ (EXMID → ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o)))
24		isomni 7128	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦 ∈ 𝑥 (𝑓‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = 1o))))
25	24	elv 2741	. . 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 1874	1 ⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834  ∀wal 1351   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  Vcvv 2737  ∅c0 3422  {cpr 3592  EXMIDwem 4191  ⟶wf 5208  ‘cfv 5212  1_oc1o 6404  2_oc2o 6405  Omnicomni 7126

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-exmid 4192  df-id 4290  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-1o 6411  df-2o 6412  df-omni 7127

This theorem is referenced by:  exmidomni  7134