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Theorem goaln0 34315
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 34264 . . . 4 𝑔𝑖𝐴 = ⟨2o, ⟨𝑖, 𝐴⟩⟩
2 2on0 8469 . . . . . . . . . . . 12 2o ≠ ∅
32neii 2943 . . . . . . . . . . 11 ¬ 2o = ∅
43intnanr 489 . . . . . . . . . 10 ¬ (2o = ∅ ∧ ⟨𝑖, 𝐴⟩ = ⟨𝑘, 𝑗⟩)
5 2oex 8464 . . . . . . . . . . 11 2o ∈ V
6 opex 5460 . . . . . . . . . . 11 𝑖, 𝐴⟩ ∈ V
75, 6opth 5472 . . . . . . . . . 10 (⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ ↔ (2o = ∅ ∧ ⟨𝑖, 𝐴⟩ = ⟨𝑘, 𝑗⟩))
84, 7mtbir 323 . . . . . . . . 9 ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩
9 goel 34269 . . . . . . . . . 10 ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘𝑔𝑗) = ⟨∅, ⟨𝑘, 𝑗⟩⟩)
109eqeq2d 2744 . . . . . . . . 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 3198 . . . . . . 7 𝑘 ∈ ω ∀𝑗 ∈ ω ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)
13 ralnex2 3134 . . . . . . 7 (∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗) ↔ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗))
1412, 13mpbi 229 . . . . . 6 ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)
1514intnan 488 . . . . 5 ¬ (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗))
16 eqeq1 2737 . . . . . . 7 (𝑥 = ⟨2o, ⟨𝑖, 𝐴⟩⟩ → (𝑥 = (𝑘𝑔𝑗) ↔ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)))
17162rexbidv 3220 . . . . . 6 (𝑥 = ⟨2o, ⟨𝑖, 𝐴⟩⟩ → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘𝑔𝑗)))
18 fmla0 34304 . . . . . 6 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘𝑔𝑗)}
1917, 18elrab2 3684 . . . . 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 2853 . . 3 ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅)
22 fveq2 6881 . . . 4 (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
2322eleq2d 2820 . . 3 (𝑁 = ∅ → (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) ↔ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅)))
2421, 23mtbiri 327 . 2 (𝑁 = ∅ → ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁))
2524necon2ai 2971 1 (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  Vcvv 3475  c0 4320  cop 4630  cfv 6535  (class class class)co 7396  ωcom 7842  2oc2o 8447  𝑔cgoe 34255  𝑔cgol 34257  Fmlacfmla 34259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-inf2 9623
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7843  df-1st 7962  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-map 8810  df-goel 34262  df-goal 34264  df-sat 34265  df-fmla 34267
This theorem is referenced by:  goalr  34319
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