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Theorem gonan0 33354
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 8318 . . . . . . . . . . . . 13 1o ≠ ∅
21neii 2945 . . . . . . . . . . . 12 ¬ 1o = ∅
32intnanr 488 . . . . . . . . . . 11 ¬ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩)
4 1oex 8307 . . . . . . . . . . . 12 1o ∈ V
5 opex 5379 . . . . . . . . . . . 12 𝐴, 𝐵⟩ ∈ V
64, 5opth 5391 . . . . . . . . . . 11 (⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩))
73, 6mtbir 323 . . . . . . . . . 10 ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩
8 goel 33309 . . . . . . . . . . 11 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
98eqeq2d 2749 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
107, 9mtbiri 327 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗))
1110rgen2 3120 . . . . . . . 8 𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)
12 ralnex2 3189 . . . . . . . 8 (∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗) ↔ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗))
1311, 12mpbi 229 . . . . . . 7 ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)
1413intnan 487 . . . . . 6 ¬ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗))
15 eqeq1 2742 . . . . . . . 8 (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (𝑥 = (𝑖𝑔𝑗) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)))
16152rexbidv 3229 . . . . . . 7 (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)))
17 fmla0 33344 . . . . . . 7 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
1816, 17elrab2 3627 . . . . . 6 (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅) ↔ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)))
1914, 18mtbir 323 . . . . 5 ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)
20 gonafv 33312 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
2120eleq1d 2823 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝑔𝐵) ∈ (Fmla‘∅) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)))
2219, 21mtbiri 327 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅))
23 eqid 2738 . . . . . . . . 9 (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
2423dmmptss 6144 . . . . . . . 8 dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) ⊆ (V × V)
25 relxp 5607 . . . . . . . 8 Rel (V × V)
26 relss 5692 . . . . . . . 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 33303 . . . . . . . . 9 𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
2928dmeqi 5813 . . . . . . . 8 dom ⊼𝑔 = dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
3029releqi 5688 . . . . . . 7 (Rel dom ⊼𝑔 ↔ Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩))
3127, 30mpbir 230 . . . . . 6 Rel dom ⊼𝑔
3231ovprc 7313 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑔𝐵) = ∅)
33 peano1 7735 . . . . . . . 8 ∅ ∈ ω
34 fmlaomn0 33352 . . . . . . . 8 (∅ ∈ ω → ∅ ∉ (Fmla‘∅))
3533, 34ax-mp 5 . . . . . . 7 ∅ ∉ (Fmla‘∅)
3635neli 3051 . . . . . 6 ¬ ∅ ∈ (Fmla‘∅)
37 eleq1 2826 . . . . . 6 ((𝐴𝑔𝐵) = ∅ → ((𝐴𝑔𝐵) ∈ (Fmla‘∅) ↔ ∅ ∈ (Fmla‘∅)))
3836, 37mtbiri 327 . . . . 5 ((𝐴𝑔𝐵) = ∅ → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅))
3932, 38syl 17 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅))
4022, 39pm2.61i 182 . . 3 ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅)
41 fveq2 6774 . . . 4 (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
4241eleq2d 2824 . . 3 (𝑁 = ∅ → ((𝐴𝑔𝐵) ∈ (Fmla‘𝑁) ↔ (𝐴𝑔𝐵) ∈ (Fmla‘∅)))
4340, 42mtbiri 327 . 2 (𝑁 = ∅ → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘𝑁))
4443necon2ai 2973 1 ((𝐴𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1539  wcel 2106  wne 2943  wnel 3049  wral 3064  wrex 3065  Vcvv 3432  wss 3887  c0 4256  cop 4567  cmpt 5157   × cxp 5587  dom cdm 5589  Rel wrel 5594  cfv 6433  (class class class)co 7275  ωcom 7712  1oc1o 8290  𝑔cgoe 33295  𝑔cgna 33296  Fmlacfmla 33299
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-map 8617  df-goel 33302  df-gona 33303  df-goal 33304  df-sat 33305  df-fmla 33307
This theorem is referenced by:  gonar  33357
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