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Theorem goaln0 32635
 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 32584 . . . 4 𝑔𝑖𝐴 = ⟨2o, ⟨𝑖, 𝐴⟩⟩
2 2on0 8107 . . . . . . . . . . . 12 2o ≠ ∅
32neii 3018 . . . . . . . . . . 11 ¬ 2o = ∅
43intnanr 490 . . . . . . . . . 10 ¬ (2o = ∅ ∧ ⟨𝑖, 𝐴⟩ = ⟨𝑘, 𝑗⟩)
5 2oex 8106 . . . . . . . . . . 11 2o ∈ V
6 opex 5348 . . . . . . . . . . 11 𝑖, 𝐴⟩ ∈ V
75, 6opth 5360 . . . . . . . . . 10 (⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ ↔ (2o = ∅ ∧ ⟨𝑖, 𝐴⟩ = ⟨𝑘, 𝑗⟩))
84, 7mtbir 325 . . . . . . . . 9 ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩
9 goel 32589 . . . . . . . . . 10 ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘𝑔𝑗) = ⟨∅, ⟨𝑘, 𝑗⟩⟩)
109eqeq2d 2832 . . . . . . . . 9 ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗) ↔ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩))
118, 10mtbiri 329 . . . . . . . 8 ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗))
1211rgen2 3203 . . . . . . 7 𝑘 ∈ ω ∀𝑗 ∈ ω ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)
13 ralnex2 3260 . . . . . . 7 (∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗) ↔ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗))
1412, 13mpbi 232 . . . . . 6 ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)
1514intnan 489 . . . . 5 ¬ (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗))
16 eqeq1 2825 . . . . . . 7 (𝑥 = ⟨2o, ⟨𝑖, 𝐴⟩⟩ → (𝑥 = (𝑘𝑔𝑗) ↔ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)))
17162rexbidv 3300 . . . . . 6 (𝑥 = ⟨2o, ⟨𝑖, 𝐴⟩⟩ → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)))
18 fmla0 32624 . . . . . 6 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘𝑔𝑗)}
1917, 18elrab2 3682 . . . . 5 (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ (Fmla‘∅) ↔ (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)))
2015, 19mtbir 325 . . . 4 ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ (Fmla‘∅)
211, 20eqneltri 2906 . . 3 ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅)
22 fveq2 6664 . . . 4 (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
2322eleq2d 2898 . . 3 (𝑁 = ∅ → (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) ↔ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅)))
2421, 23mtbiri 329 . 2 (𝑁 = ∅ → ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁))
2524necon2ai 3045 1 (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494  ∅c0 4290  ⟨cop 4566  ‘cfv 6349  (class class class)co 7150  ωcom 7574  2oc2o 8090  ∈𝑔cgoe 32575  ∀𝑔cgol 32577  Fmlacfmla 32579 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-map 8402  df-goel 32582  df-goal 32584  df-sat 32585  df-fmla 32587 This theorem is referenced by:  goalr  32639
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