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	⊢ ((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 8150	. . . . . . . . . . . . 13 ⊢ 1o ≠ ∅
2	1	neii 2936	. . . . . . . . . . . 12 ⊢ ¬ 1o = ∅
3	2	intnanr 491	. . . . . . . . . . 11 ⊢ ¬ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩)
4		1oex 8144	. . . . . . . . . . . 12 ⊢ 1o ∈ V
5		opex 5322	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
6	4, 5	opth 5334	. . . . . . . . . . 11 ⊢ (⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩))
7	3, 6	mtbir 326	. . . . . . . . . 10 ⊢ ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩
8		goel 32880	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
9	8	eqeq2d 2749	. . . . . . . . . 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 3115	. . . . . . . 8 ⊢ ∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)
12		ralnex2 3172	. . . . . . . 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 490	. . . . . 6 ⊢ ¬ (⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗))
15		eqeq1 2742	. . . . . . . 8 ⊢ (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (𝑥 = (𝑖∈𝑔𝑗) ↔ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)))
16	15	2rexbidv 3210	. . . . . . 7 ⊢ (𝑥 = ⟨1o, ⟨𝐴, 𝐵⟩⟩ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)))
17		fmla0 32915	. . . . . . 7 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
18	16, 17	elrab2 3591	. . . . . 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 32883	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴⊼𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
21	20	eleq1d 2817	. . . . 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 2738	. . . . . . . . 9 ⊢ (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
24	23	dmmptss 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⟩)))
27	24, 25, 26	mp2 9	. . . . . . 7 ⊢ Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
28		df-gona 32874	. . . . . . . . 9 ⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
29	28	dmeqi 5747	. . . . . . . 8 ⊢ dom ⊼𝑔 = dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
30	29	releqi 5623	. . . . . . 7 ⊢ (Rel dom ⊼𝑔 ↔ Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩))
31	27, 30	mpbir 234	. . . . . 6 ⊢ Rel dom ⊼𝑔
32	31	ovprc 7208	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴⊼𝑔𝐵) = ∅)
33		peano1 7620	. . . . . . . 8 ⊢ ∅ ∈ ω
34		fmlaomn0 32923	. . . . . . . 8 ⊢ (∅ ∈ ω → ∅ ∉ (Fmla‘∅))
35	33, 34	ax-mp 5	. . . . . . 7 ⊢ ∅ ∉ (Fmla‘∅)
36	35	neli 3040	. . . . . 6 ⊢ ¬ ∅ ∈ (Fmla‘∅)
37		eleq1 2820	. . . . . 6 ⊢ ((𝐴⊼𝑔𝐵) = ∅ → ((𝐴⊼𝑔𝐵) ∈ (Fmla‘∅) ↔ ∅ ∈ (Fmla‘∅)))
38	36, 37	mtbiri 330	. . . . 5 ⊢ ((𝐴⊼𝑔𝐵) = ∅ → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅))
39	32, 38	syl 17	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅))
40	22, 39	pm2.61i 185	. . 3 ⊢ ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅)
41		fveq2 6674	. . . 4 ⊢ (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
42	41	eleq2d 2818	. . 3 ⊢ (𝑁 = ∅ → ((𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁) ↔ (𝐴⊼𝑔𝐵) ∈ (Fmla‘∅)))
43	40, 42	mtbiri 330	. 2 ⊢ (𝑁 = ∅ → ¬ (𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁))
44	43	necon2ai 2963	1 ⊢ ((𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)

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  1_oc1o 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