Theorem gonan0 35590

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	⊢ ((𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)

Proof of Theorem gonan0

Dummy variables 𝑖 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 8416	. . . . . . . . . . . . 13 ⊢ 1o ≠ ∅
2	1	neii 2935	. . . . . . . . . . . 12 ⊢ ¬ 1o = ∅
3	2	intnanr 487	. . . . . . . . . . 11 ⊢ ¬ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩)
4		1oex 8408	. . . . . . . . . . . 12 ⊢ 1o ∈ V
5		opex 5411	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
6	4, 5	opth 5424	. . . . . . . . . . 11 ⊢ (⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩))
7	3, 6	mtbir 323	. . . . . . . . . 10 ⊢ ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩
8		goel 35545	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
9	8	eqeq2d 2748	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
10	7, 9	mtbiri 327	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗))
11	10	rgen2 3178	. . . . . . . 8 ⊢ ∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)
12		ralnex2 3118	. . . . . . . 8 ⊢ (∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗) ↔ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗))
13	11, 12	mpbi 230	. . . . . . 7 ⊢ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)
14	13	intnan 486	. . . . . 6 ⊢ ¬ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗))
15		eqeq1 2741	. . . . . . . 8 ⊢ (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (𝑥 = (𝑖∈𝑔𝑗) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)))
16	15	2rexbidv 3203	. . . . . . 7 ⊢ (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)))
17		fmla0 35580	. . . . . . 7 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
18	16, 17	elrab2 3638	. . . . . 6 ⊢ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅) ↔ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)))
19	14, 18	mtbir 323	. . . . 5 ⊢ ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)
20		gonafv 35548	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴⊼𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
21	20	eleq1d 2822	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴⊼𝑔𝐵) ∈ (Fmla‘∅) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)))
22	19, 21	mtbiri 327	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅))
23		eqid 2737	. . . . . . . . 9 ⊢ (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
24	23	dmmptss 6199	. . . . . . . 8 ⊢ dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) ⊆ (V × V)
25		relxp 5642	. . . . . . . 8 ⊢ Rel (V × V)
26		relss 5731	. . . . . . . 8 ⊢ (dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) ⊆ (V × V) → (Rel (V × V) → Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)))
27	24, 25, 26	mp2 9	. . . . . . 7 ⊢ Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
28		df-gona 35539	. . . . . . . . 9 ⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
29	28	dmeqi 5853	. . . . . . . 8 ⊢ dom ⊼𝑔 = dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
30	29	releqi 5727	. . . . . . 7 ⊢ (Rel dom ⊼𝑔 ↔ Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩))
31	27, 30	mpbir 231	. . . . . 6 ⊢ Rel dom ⊼𝑔
32	31	ovprc 7398	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴⊼𝑔𝐵) = ∅)
33		peano1 7833	. . . . . . . 8 ⊢ ∅ ∈ ω
34		fmlaomn0 35588	. . . . . . . 8 ⊢ (∅ ∈ ω → ∅ ∉ (Fmla‘∅))
35	33, 34	ax-mp 5	. . . . . . 7 ⊢ ∅ ∉ (Fmla‘∅)
36	35	neli 3039	. . . . . 6 ⊢ ¬ ∅ ∈ (Fmla‘∅)
37		eleq1 2825	. . . . . 6 ⊢ ((𝐴⊼𝑔𝐵) = ∅ → ((𝐴⊼𝑔𝐵) ∈ (Fmla‘∅) ↔ ∅ ∈ (Fmla‘∅)))
38	36, 37	mtbiri 327	. . . . 5 ⊢ ((𝐴⊼𝑔𝐵) = ∅ → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅))
39	32, 38	syl 17	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅))
40	22, 39	pm2.61i 182	. . 3 ⊢ ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅)
41		fveq2 6834	. . . 4 ⊢ (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
42	41	eleq2d 2823	. . 3 ⊢ (𝑁 = ∅ → ((𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁) ↔ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅)))
43	40, 42	mtbiri 327	. 2 ⊢ (𝑁 = ∅ → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁))
44	43	necon2ai 2962	1 ⊢ ((𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∉ wnel 3037  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  ⟨cop 4574   ↦ cmpt 5167   × cxp 5622  dom cdm 5624  Rel wrel 5629  ‘cfv 6492  (class class class)co 7360  ωcom 7810  1_oc1o 8391  ∈_𝑔cgoe 35531  ⊼_𝑔cgna 35532  Fmlacfmla 35535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-map 8768  df-goel 35538  df-gona 35539  df-goal 35540  df-sat 35541  df-fmla 35543

This theorem is referenced by:  gonar  35593