Theorem goaln0 35376

Description: The "Godel-set of universal quantification" is a Godel formula of at least height 1. (Contributed by AV, 22-Oct-2023.)

Assertion

Ref	Expression
goaln0	⊢ (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)

Distinct variable group:   𝐴,𝑖

Allowed substitution hint:   𝑁(𝑖)

Proof of Theorem goaln0

Dummy variables 𝑗 𝑥 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-goal 35325	. . . 4 ⊢ ∀𝑔𝑖𝐴 = ⟨2o, ⟨𝑖, 𝐴⟩⟩
2		2on0 8402	. . . . . . . . . . . 12 ⊢ 2o ≠ ∅
3	2	neii 2927	. . . . . . . . . . 11 ⊢ ¬ 2o = ∅
4	3	intnanr 487	. . . . . . . . . 10 ⊢ ¬ (2o = ∅ ∧ ⟨𝑖, 𝐴⟩ = ⟨𝑘, 𝑗⟩)
5		2oex 8399	. . . . . . . . . . 11 ⊢ 2o ∈ V
6		opex 5407	. . . . . . . . . . 11 ⊢ ⟨𝑖, 𝐴⟩ ∈ V
7	5, 6	opth 5419	. . . . . . . . . 10 ⊢ (⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ ↔ (2o = ∅ ∧ ⟨𝑖, 𝐴⟩ = ⟨𝑘, 𝑗⟩))
8	4, 7	mtbir 323	. . . . . . . . 9 ⊢ ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩
9		goel 35330	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘∈𝑔𝑗) = ⟨∅, ⟨𝑘, 𝑗⟩⟩)
10	9	eqeq2d 2740	. . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗) ↔ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩))
11	8, 10	mtbiri 327	. . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗))
12	11	rgen2 3169	. . . . . . 7 ⊢ ∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗)
13		ralnex2 3109	. . . . . . 7 ⊢ (∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗) ↔ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗))
14	12, 13	mpbi 230	. . . . . 6 ⊢ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗)
15	14	intnan 486	. . . . 5 ⊢ ¬ (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗))
16		eqeq1 2733	. . . . . . 7 ⊢ (𝑥 = ⟨2o, ⟨𝑖, 𝐴⟩⟩ → (𝑥 = (𝑘∈𝑔𝑗) ↔ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗)))
17	16	2rexbidv 3194	. . . . . 6 ⊢ (𝑥 = ⟨2o, ⟨𝑖, 𝐴⟩⟩ → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗)))
18		fmla0 35365	. . . . . 6 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗)}
19	17, 18	elrab2 3651	. . . . 5 ⊢ (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ (Fmla‘∅) ↔ (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗)))
20	15, 19	mtbir 323	. . . 4 ⊢ ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ (Fmla‘∅)
21	1, 20	eqneltri 2847	. . 3 ⊢ ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅)
22		fveq2 6822	. . . 4 ⊢ (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
23	22	eleq2d 2814	. . 3 ⊢ (𝑁 = ∅ → (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) ↔ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅)))
24	21, 23	mtbiri 327	. 2 ⊢ (𝑁 = ∅ → ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁))
25	24	necon2ai 2954	1 ⊢ (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3436  ∅c0 4284  ⟨cop 4583  ‘cfv 6482  (class class class)co 7349  ωcom 7799  2_oc2o 8382  ∈_𝑔cgoe 35316  ∀_𝑔cgol 35318  Fmlacfmla 35320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-map 8755  df-goel 35323  df-goal 35325  df-sat 35326  df-fmla 35328

This theorem is referenced by:  goalr  35380