Theorem gonan0 35865

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 8473	. . . . . . . . . . . . 13 ⊢ 1o ≠ ∅
2	1	neii 2960	. . . . . . . . . . . 12 ⊢ ¬ 1o = ∅
3	2	intnanr 492	. . . . . . . . . . 11 ⊢ ¬ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩)
4		1oex 8464	. . . . . . . . . . . 12 ⊢ 1o ∈ V
5		opex 5447	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
6	4, 5	opth 5460	. . . . . . . . . . 11 ⊢ (⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩))
7	3, 6	mtbir 326	. . . . . . . . . 10 ⊢ ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩
8		goel 35820	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
9	8	eqeq2d 2774	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
10	7, 9	mtbiri 330	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗))
11	10	rgen2 3205	. . . . . . . 8 ⊢ ∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)
12		ralnex2 3145	. . . . . . . 8 ⊢ (∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗) ↔ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗))
13	11, 12	mpbi 233	. . . . . . 7 ⊢ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)
14	13	intnan 491	. . . . . 6 ⊢ ¬ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗))
15		eqeq1 2767	. . . . . . . 8 ⊢ (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (𝑥 = (𝑖∈𝑔𝑗) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)))
16	15	2rexbidv 3230	. . . . . . 7 ⊢ (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)))
17		fmla0 35855	. . . . . . 7 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
18	16, 17	elrab2 3655	. . . . . 6 ⊢ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅) ↔ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)))
19	14, 18	mtbir 326	. . . . 5 ⊢ ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)
20		gonafv 35823	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴⊼𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
21	20	eleq1d 2848	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴⊼𝑔𝐵) ∈ (Fmla‘∅) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ (Fmla‘∅)))
22	19, 21	mtbiri 330	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅))
23		eqid 2763	. . . . . . . . 9 ⊢ (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
24	23	dmmptss 6244	. . . . . . . 8 ⊢ dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) ⊆ (V × V)
25		relxp 5681	. . . . . . . 8 ⊢ Rel (V × V)
26		relss 5770	. . . . . . . 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 35814	. . . . . . . . 9 ⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
29	28	dmeqi 5896	. . . . . . . 8 ⊢ dom ⊼𝑔 = dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
30	29	releqi 5766	. . . . . . 7 ⊢ (Rel dom ⊼𝑔 ↔ Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩))
31	27, 30	mpbir 234	. . . . . 6 ⊢ Rel dom ⊼𝑔
32	31	ovprc 7450	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴⊼𝑔𝐵) = ∅)
33		peano1 7886	. . . . . . . 8 ⊢ ∅ ∈ ω
34		fmlaomn0 35863	. . . . . . . 8 ⊢ (∅ ∈ ω → ∅ ∉ (Fmla‘∅))
35	33, 34	ax-mp 5	. . . . . . 7 ⊢ ∅ ∉ (Fmla‘∅)
36	35	neli 3066	. . . . . 6 ⊢ ¬ ∅ ∈ (Fmla‘∅)
37		eleq1 2851	. . . . . 6 ⊢ ((𝐴⊼𝑔𝐵) = ∅ → ((𝐴⊼𝑔𝐵) ∈ (Fmla‘∅) ↔ ∅ ∈ (Fmla‘∅)))
38	36, 37	mtbiri 330	. . . . 5 ⊢ ((𝐴⊼𝑔𝐵) = ∅ → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅))
39	32, 38	syl 18	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅))
40	22, 39	pm2.61i 184	. . 3 ⊢ ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅)
41		fveq2 6883	. . . 4 ⊢ (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
42	41	eleq2d 2849	. . 3 ⊢ (𝑁 = ∅ → ((𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁) ↔ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅)))
43	40, 42	mtbiri 330	. 2 ⊢ (𝑁 = ∅ → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁))
44	43	necon2ai 2987	1 ⊢ ((𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   ∉ wnel 3064  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ⊆ wss 3906  ∅c0 4287  ⟨cop 4596   ↦ cmpt 5193   × cxp 5661  dom cdm 5663  Rel wrel 5668  ‘cfv 6538  (class class class)co 7412  ωcom 7863  1_oc1o 8447  ∈_𝑔cgoe 35806  ⊼_𝑔cgna 35807  Fmlacfmla 35810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-inf2 9611

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-map 8827  df-goel 35813  df-gona 35814  df-goal 35815  df-sat 35816  df-fmla 35818

This theorem is referenced by:  gonar  35868