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Theorem gonan0 35377
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 8525 . . . . . . . . . . . . 13 1o ≠ ∅
21neii 2940 . . . . . . . . . . . 12 ¬ 1o = ∅
32intnanr 487 . . . . . . . . . . 11 ¬ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩)
4 1oex 8515 . . . . . . . . . . . 12 1o ∈ V
5 opex 5475 . . . . . . . . . . . 12 𝐴, 𝐵⟩ ∈ V
64, 5opth 5487 . . . . . . . . . . 11 (⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩))
73, 6mtbir 323 . . . . . . . . . 10 ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩
8 goel 35332 . . . . . . . . . . 11 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
98eqeq2d 2746 . . . . . . . . . 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 3197 . . . . . . . 8 𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)
12 ralnex2 3131 . . . . . . . 8 (∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗) ↔ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗))
1311, 12mpbi 230 . . . . . . 7 ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)
1413intnan 486 . . . . . 6 ¬ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗))
15 eqeq1 2739 . . . . . . . 8 (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (𝑥 = (𝑖𝑔𝑗) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)))
16152rexbidv 3220 . . . . . . 7 (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)))
17 fmla0 35367 . . . . . . 7 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
1816, 17elrab2 3698 . . . . . 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 35335 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
2120eleq1d 2824 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝑔𝐵) ∈ (Fmla‘∅) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)))
2219, 21mtbiri 327 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅))
23 eqid 2735 . . . . . . . . 9 (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
2423dmmptss 6263 . . . . . . . 8 dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) ⊆ (V × V)
25 relxp 5707 . . . . . . . 8 Rel (V × V)
26 relss 5794 . . . . . . . 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 35326 . . . . . . . . 9 𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
2928dmeqi 5918 . . . . . . . 8 dom ⊼𝑔 = dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
3029releqi 5790 . . . . . . 7 (Rel dom ⊼𝑔 ↔ Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩))
3127, 30mpbir 231 . . . . . 6 Rel dom ⊼𝑔
3231ovprc 7469 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑔𝐵) = ∅)
33 peano1 7911 . . . . . . . 8 ∅ ∈ ω
34 fmlaomn0 35375 . . . . . . . 8 (∅ ∈ ω → ∅ ∉ (Fmla‘∅))
3533, 34ax-mp 5 . . . . . . 7 ∅ ∉ (Fmla‘∅)
3635neli 3046 . . . . . 6 ¬ ∅ ∈ (Fmla‘∅)
37 eleq1 2827 . . . . . 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 6907 . . . 4 (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
4241eleq2d 2825 . . 3 (𝑁 = ∅ → ((𝐴𝑔𝐵) ∈ (Fmla‘𝑁) ↔ (𝐴𝑔𝐵) ∈ (Fmla‘∅)))
4340, 42mtbiri 327 . 2 (𝑁 = ∅ → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘𝑁))
4443necon2ai 2968 1 ((𝐴𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  wnel 3044  wral 3059  wrex 3068  Vcvv 3478  wss 3963  c0 4339  cop 4637  cmpt 5231   × cxp 5687  dom cdm 5689  Rel wrel 5694  cfv 6563  (class class class)co 7431  ωcom 7887  1oc1o 8498  𝑔cgoe 35318  𝑔cgna 35319  Fmlacfmla 35322
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-map 8867  df-goel 35325  df-gona 35326  df-goal 35327  df-sat 35328  df-fmla 35330
This theorem is referenced by:  gonar  35380
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