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Theorem gonan0 32925
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 8150 . . . . . . . . . . . . 13 1o ≠ ∅
21neii 2936 . . . . . . . . . . . 12 ¬ 1o = ∅
32intnanr 491 . . . . . . . . . . 11 ¬ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩)
4 1oex 8144 . . . . . . . . . . . 12 1o ∈ V
5 opex 5322 . . . . . . . . . . . 12 𝐴, 𝐵⟩ ∈ V
64, 5opth 5334 . . . . . . . . . . 11 (⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩))
73, 6mtbir 326 . . . . . . . . . 10 ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩
8 goel 32880 . . . . . . . . . . 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 330 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗))
1110rgen2 3115 . . . . . . . 8 𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)
12 ralnex2 3172 . . . . . . . 8 (∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗) ↔ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗))
1311, 12mpbi 233 . . . . . . 7 ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)
1413intnan 490 . . . . . 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 3210 . . . . . . 7 (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)))
17 fmla0 32915 . . . . . . 7 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
1816, 17elrab2 3591 . . . . . 6 (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅) ↔ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖𝑔𝑗)))
1914, 18mtbir 326 . . . . 5 ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)
20 gonafv 32883 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
2120eleq1d 2817 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝑔𝐵) ∈ (Fmla‘∅) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)))
2219, 21mtbiri 330 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅))
23 eqid 2738 . . . . . . . . 9 (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
2423dmmptss 6073 . . . . . . . 8 dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) ⊆ (V × V)
25 relxp 5543 . . . . . . . 8 Rel (V × V)
26 relss 5627 . . . . . . . 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 32874 . . . . . . . . 9 𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
2928dmeqi 5747 . . . . . . . 8 dom ⊼𝑔 = dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
3029releqi 5623 . . . . . . 7 (Rel dom ⊼𝑔 ↔ Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩))
3127, 30mpbir 234 . . . . . 6 Rel dom ⊼𝑔
3231ovprc 7208 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑔𝐵) = ∅)
33 peano1 7620 . . . . . . . 8 ∅ ∈ ω
34 fmlaomn0 32923 . . . . . . . 8 (∅ ∈ ω → ∅ ∉ (Fmla‘∅))
3533, 34ax-mp 5 . . . . . . 7 ∅ ∉ (Fmla‘∅)
3635neli 3040 . . . . . 6 ¬ ∅ ∈ (Fmla‘∅)
37 eleq1 2820 . . . . . 6 ((𝐴𝑔𝐵) = ∅ → ((𝐴𝑔𝐵) ∈ (Fmla‘∅) ↔ ∅ ∈ (Fmla‘∅)))
3836, 37mtbiri 330 . . . . 5 ((𝐴𝑔𝐵) = ∅ → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅))
3932, 38syl 17 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅))
4022, 39pm2.61i 185 . . 3 ¬ (𝐴𝑔𝐵) ∈ (Fmla‘∅)
41 fveq2 6674 . . . 4 (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
4241eleq2d 2818 . . 3 (𝑁 = ∅ → ((𝐴𝑔𝐵) ∈ (Fmla‘𝑁) ↔ (𝐴𝑔𝐵) ∈ (Fmla‘∅)))
4340, 42mtbiri 330 . 2 (𝑁 = ∅ → ¬ (𝐴𝑔𝐵) ∈ (Fmla‘𝑁))
4443necon2ai 2963 1 ((𝐴𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1542  wcel 2114  wne 2934  wnel 3038  wral 3053  wrex 3054  Vcvv 3398  wss 3843  c0 4211  cop 4522  cmpt 5110   × cxp 5523  dom cdm 5525  Rel wrel 5530  cfv 6339  (class class class)co 7170  ωcom 7599  1oc1o 8124  𝑔cgoe 32866  𝑔cgna 32867  Fmlacfmla 32870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-2o 8132  df-map 8439  df-goel 32873  df-gona 32874  df-goal 32875  df-sat 32876  df-fmla 32878
This theorem is referenced by:  gonar  32928
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