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Theorem gonan0 32634
Description: The "Godel-set of NAND" is a Godel formula of at least height 1. (Contributed by AV, 21-Oct-2023.)
Assertion
Ref Expression
gonan0 ((𝐴𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)

Proof of Theorem gonan0
Dummy variables 𝑖 𝑗 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 8113 . . . . . . . . . . . . 13 1o ≠ ∅
21neii 3018 . . . . . . . . . . . 12 ¬ 1o = ∅
32intnanr 490 . . . . . . . . . . 11 ¬ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩)
4 1oex 8104 . . . . . . . . . . . 12 1o ∈ V
5 opex 5348 . . . . . . . . . . . 12 𝐴, 𝐵⟩ ∈ V
64, 5opth 5360 . . . . . . . . . . 11 (⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩))
73, 6mtbir 325 . . . . . . . . . 10 ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩
8 goel 32589 . . . . . . . . . . 11 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
98eqeq2d 2832 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
107, 9mtbiri 329 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗))
1110rgen2 3203 . . . . . . . 8 𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)
12 ralnex2 3260 . . . . . . . 8 (∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗) ↔ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗))
1311, 12mpbi 232 . . . . . . 7 ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)
1413intnan 489 . . . . . 6 ¬ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗))
15 eqeq1 2825 . . . . . . . 8 (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (𝑥 = (𝑖𝑔𝑗) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)))
16152rexbidv 3300 . . . . . . 7 (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)))
17 fmla0 32624 . . . . . . 7 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
1816, 17elrab2 3682 . . . . . 6 (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅) ↔ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)))
1914, 18mtbir 325 . . . . 5 ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)
20 gonafv 32592 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
2120eleq1d 2897 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝑔𝐵) ∈ (Fmla‘∅) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)))
2219, 21mtbiri 329 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅))
23 eqid 2821 . . . . . . . . 9 (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
2423dmmptss 6089 . . . . . . . 8 dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) ⊆ (V × V)
25 relxp 5567 . . . . . . . 8 Rel (V × V)
26 relss 5650 . . . . . . . 8 (dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) ⊆ (V × V) → (Rel (V × V) → Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)))
2724, 25, 26mp2 9 . . . . . . 7 Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
28 df-gona 32583 . . . . . . . . 9 𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
2928dmeqi 5767 . . . . . . . 8 dom ⊼𝑔 = dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
3029releqi 5646 . . . . . . 7 (Rel dom ⊼𝑔 ↔ Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩))
3127, 30mpbir 233 . . . . . 6 Rel dom ⊼𝑔
3231ovprc 7188 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑔𝐵) = ∅)
33 peano1 7595 . . . . . . . 8 ∅ ∈ ω
34 fmlaomn0 32632 . . . . . . . 8 (∅ ∈ ω → ∅ ∉ (Fmla‘∅))
3533, 34ax-mp 5 . . . . . . 7 ∅ ∉ (Fmla‘∅)
3635neli 3125 . . . . . 6 ¬ ∅ ∈ (Fmla‘∅)
37 eleq1 2900 . . . . . 6 ((𝐴𝑔𝐵) = ∅ → ((𝐴𝑔𝐵) ∈ (Fmla‘∅) ↔ ∅ ∈ (Fmla‘∅)))
3836, 37mtbiri 329 . . . . 5 ((𝐴𝑔𝐵) = ∅ → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅))
3932, 38syl 17 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅))
4022, 39pm2.61i 184 . . 3 ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅)
41 fveq2 6664 . . . 4 (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
4241eleq2d 2898 . . 3 (𝑁 = ∅ → ((𝐴𝑔𝐵) ∈ (Fmla‘𝑁) ↔ (𝐴𝑔𝐵) ∈ (Fmla‘∅)))
4340, 42mtbiri 329 . 2 (𝑁 = ∅ → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘𝑁))
4443necon2ai 3045 1 ((𝐴𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1533  wcel 2110  wne 3016  wnel 3123  wral 3138  wrex 3139  Vcvv 3494  wss 3935  c0 4290  cop 4566  cmpt 5138   × cxp 5547  dom cdm 5549  Rel wrel 5554  cfv 6349  (class class class)co 7150  ωcom 7574  1oc1o 8089  𝑔cgoe 32575  𝑔cgna 32576  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-nel 3124  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-gona 32583  df-goal 32584  df-sat 32585  df-fmla 32587
This theorem is referenced by:  gonar  32637
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