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Theorem goaln0 35398
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.
StepHypRef Expression
1 df-goal 35347 . . . 4 𝑔𝑖𝐴 = ⟨2o, ⟨𝑖, 𝐴⟩⟩
2 2on0 8522 . . . . . . . . . . . 12 2o ≠ ∅
32neii 2942 . . . . . . . . . . 11 ¬ 2o = ∅
43intnanr 487 . . . . . . . . . 10 ¬ (2o = ∅ ∧ ⟨𝑖, 𝐴⟩ = ⟨𝑘, 𝑗⟩)
5 2oex 8517 . . . . . . . . . . 11 2o ∈ V
6 opex 5469 . . . . . . . . . . 11 𝑖, 𝐴⟩ ∈ V
75, 6opth 5481 . . . . . . . . . 10 (⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ ↔ (2o = ∅ ∧ ⟨𝑖, 𝐴⟩ = ⟨𝑘, 𝑗⟩))
84, 7mtbir 323 . . . . . . . . 9 ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩
9 goel 35352 . . . . . . . . . 10 ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘𝑔𝑗) = ⟨∅, ⟨𝑘, 𝑗⟩⟩)
109eqeq2d 2748 . . . . . . . . 9 ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗) ↔ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩))
118, 10mtbiri 327 . . . . . . . 8 ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗))
1211rgen2 3199 . . . . . . 7 𝑘 ∈ ω ∀𝑗 ∈ ω ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)
13 ralnex2 3133 . . . . . . 7 (∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗) ↔ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗))
1412, 13mpbi 230 . . . . . 6 ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)
1514intnan 486 . . . . 5 ¬ (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗))
16 eqeq1 2741 . . . . . . 7 (𝑥 = ⟨2o, ⟨𝑖, 𝐴⟩⟩ → (𝑥 = (𝑘𝑔𝑗) ↔ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)))
17162rexbidv 3222 . . . . . 6 (𝑥 = ⟨2o, ⟨𝑖, 𝐴⟩⟩ → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)))
18 fmla0 35387 . . . . . 6 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘𝑔𝑗)}
1917, 18elrab2 3695 . . . . 5 (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ (Fmla‘∅) ↔ (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)))
2015, 19mtbir 323 . . . 4 ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ (Fmla‘∅)
211, 20eqneltri 2860 . . 3 ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅)
22 fveq2 6906 . . . 4 (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
2322eleq2d 2827 . . 3 (𝑁 = ∅ → (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) ↔ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅)))
2421, 23mtbiri 327 . 2 (𝑁 = ∅ → ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁))
2524necon2ai 2970 1 (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  c0 4333  cop 4632  cfv 6561  (class class class)co 7431  ωcom 7887  2oc2o 8500  𝑔cgoe 35338  𝑔cgol 35340  Fmlacfmla 35342
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-map 8868  df-goel 35345  df-goal 35347  df-sat 35348  df-fmla 35350
This theorem is referenced by:  goalr  35402
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