Theorem gonan0 32639

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 8119	. . . . . . . . . . . . 13 ⊢ 1o ≠ ∅
2	1	neii 3018	. . . . . . . . . . . 12 ⊢ ¬ 1o = ∅
3	2	intnanr 490	. . . . . . . . . . 11 ⊢ ¬ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩)
4		1oex 8110	. . . . . . . . . . . 12 ⊢ 1o ∈ V
5		opex 5356	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
6	4, 5	opth 5368	. . . . . . . . . . 11 ⊢ (⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩))
7	3, 6	mtbir 325	. . . . . . . . . 10 ⊢ ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩
8		goel 32594	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
9	8	eqeq2d 2832	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
10	7, 9	mtbiri 329	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗))
11	10	rgen2 3203	. . . . . . . 8 ⊢ ∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)
12		ralnex2 3260	. . . . . . . 8 ⊢ (∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗) ↔ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗))
13	11, 12	mpbi 232	. . . . . . 7 ⊢ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)
14	13	intnan 489	. . . . . 6 ⊢ ¬ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗))
15		eqeq1 2825	. . . . . . . 8 ⊢ (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (𝑥 = (𝑖∈𝑔𝑗) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)))
16	15	2rexbidv 3300	. . . . . . 7 ⊢ (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)))
17		fmla0 32629	. . . . . . 7 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
18	16, 17	elrab2 3683	. . . . . 6 ⊢ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅) ↔ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)))
19	14, 18	mtbir 325	. . . . 5 ⊢ ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)
20		gonafv 32597	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴⊼𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
21	20	eleq1d 2897	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴⊼𝑔𝐵) ∈ (Fmla‘∅) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)))
22	19, 21	mtbiri 329	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅))
23		eqid 2821	. . . . . . . . 9 ⊢ (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
24	23	dmmptss 6095	. . . . . . . 8 ⊢ dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) ⊆ (V × V)
25		relxp 5573	. . . . . . . 8 ⊢ Rel (V × V)
26		relss 5656	. . . . . . . 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 32588	. . . . . . . . 9 ⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
29	28	dmeqi 5773	. . . . . . . 8 ⊢ dom ⊼𝑔 = dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
30	29	releqi 5652	. . . . . . 7 ⊢ (Rel dom ⊼𝑔 ↔ Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩))
31	27, 30	mpbir 233	. . . . . 6 ⊢ Rel dom ⊼𝑔
32	31	ovprc 7194	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴⊼𝑔𝐵) = ∅)
33		peano1 7601	. . . . . . . 8 ⊢ ∅ ∈ ω
34		fmlaomn0 32637	. . . . . . . 8 ⊢ (∅ ∈ ω → ∅ ∉ (Fmla‘∅))
35	33, 34	ax-mp 5	. . . . . . 7 ⊢ ∅ ∉ (Fmla‘∅)
36	35	neli 3125	. . . . . 6 ⊢ ¬ ∅ ∈ (Fmla‘∅)
37		eleq1 2900	. . . . . 6 ⊢ ((𝐴⊼𝑔𝐵) = ∅ → ((𝐴⊼𝑔𝐵) ∈ (Fmla‘∅) ↔ ∅ ∈ (Fmla‘∅)))
38	36, 37	mtbiri 329	. . . . 5 ⊢ ((𝐴⊼𝑔𝐵) = ∅ → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅))
39	32, 38	syl 17	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅))
40	22, 39	pm2.61i 184	. . 3 ⊢ ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅)
41		fveq2 6670	. . . 4 ⊢ (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
42	41	eleq2d 2898	. . 3 ⊢ (𝑁 = ∅ → ((𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁) ↔ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅)))
43	40, 42	mtbiri 329	. 2 ⊢ (𝑁 = ∅ → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁))
44	43	necon2ai 3045	1 ⊢ ((𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016   ∉ wnel 3123  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ⊆ wss 3936  ∅c0 4291  ⟨cop 4573   ↦ cmpt 5146   × cxp 5553  dom cdm 5555  Rel wrel 5560  ‘cfv 6355  (class class class)co 7156  ωcom 7580  1_oc1o 8095  ∈_𝑔cgoe 32580  ⊼_𝑔cgna 32581  Fmlacfmla 32584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-map 8408  df-goel 32587  df-gona 32588  df-goal 32589  df-sat 32590  df-fmla 32592

This theorem is referenced by:  gonar  32642