Theorem gonan0 35419

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 8505	. . . . . . . . . . . . 13 ⊢ 1o ≠ ∅
2	1	neii 2935	. . . . . . . . . . . 12 ⊢ ¬ 1o = ∅
3	2	intnanr 487	. . . . . . . . . . 11 ⊢ ¬ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩)
4		1oex 8495	. . . . . . . . . . . 12 ⊢ 1o ∈ V
5		opex 5444	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
6	4, 5	opth 5456	. . . . . . . . . . 11 ⊢ (⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (1o = ∅ ∧ ⟨𝐴, 𝐵⟩ = ⟨𝑖, 𝑗⟩))
7	3, 6	mtbir 323	. . . . . . . . . 10 ⊢ ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩
8		goel 35374	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
9	8	eqeq2d 2747	. . . . . . . . . 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 3185	. . . . . . . 8 ⊢ ∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ⟨1o, ⟨𝐴, 𝐵⟩⟩ = (𝑖∈𝑔𝑗)
12		ralnex2 3121	. . . . . . . 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 2740	. . . . . . . 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 35409	. . . . . . 7 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
18	16, 17	elrab2 3679	. . . . . 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 35377	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴⊼𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
21	20	eleq1d 2820	. . . . 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 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
24	23	dmmptss 6235	. . . . . . . 8 ⊢ dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩) ⊆ (V × V)
25		relxp 5677	. . . . . . . 8 ⊢ Rel (V × V)
26		relss 5765	. . . . . . . 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 35368	. . . . . . . . 9 ⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
29	28	dmeqi 5889	. . . . . . . 8 ⊢ dom ⊼𝑔 = dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
30	29	releqi 5761	. . . . . . 7 ⊢ (Rel dom ⊼𝑔 ↔ Rel dom (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩))
31	27, 30	mpbir 231	. . . . . 6 ⊢ Rel dom ⊼𝑔
32	31	ovprc 7448	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴⊼𝑔𝐵) = ∅)
33		peano1 7889	. . . . . . . 8 ⊢ ∅ ∈ ω
34		fmlaomn0 35417	. . . . . . . 8 ⊢ (∅ ∈ ω → ∅ ∉ (Fmla‘∅))
35	33, 34	ax-mp 5	. . . . . . 7 ⊢ ∅ ∉ (Fmla‘∅)
36	35	neli 3039	. . . . . 6 ⊢ ¬ ∅ ∈ (Fmla‘∅)
37		eleq1 2823	. . . . . 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 6881	. . . 4 ⊢ (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
42	41	eleq2d 2821	. . 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 1540   ∈ wcel 2109   ≠ wne 2933   ∉ wnel 3037  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ⊆ wss 3931  ∅c0 4313  ⟨cop 4612   ↦ cmpt 5206   × cxp 5657  dom cdm 5659  Rel wrel 5664  ‘cfv 6536  (class class class)co 7410  ωcom 7866  1_oc1o 8478  ∈_𝑔cgoe 35360  ⊼_𝑔cgna 35361  Fmlacfmla 35364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-map 8847  df-goel 35367  df-gona 35368  df-goal 35369  df-sat 35370  df-fmla 35372

This theorem is referenced by:  gonar  35422