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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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