Theorem goaln0 35589

Description: The "Godel-set of universal quantification" is a Godel formula of at least height 1. (Contributed by AV, 22-Oct-2023.)

Assertion

Ref	Expression
goaln0	⊢ (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)

Distinct variable group:   𝐴,𝑖

Allowed substitution hint:   𝑁(𝑖)

Proof of Theorem goaln0

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

Step	Hyp	Ref	Expression
1		df-goal 35538	. . . 4 ⊢ ∀𝑔𝑖𝐴 = ⟨2o, ⟨𝑖, 𝐴⟩⟩
2		2on0 8411	. . . . . . . . . . . 12 ⊢ 2o ≠ ∅
3	2	neii 2934	. . . . . . . . . . 11 ⊢ ¬ 2o = ∅
4	3	intnanr 487	. . . . . . . . . 10 ⊢ ¬ (2o = ∅ ∧ ⟨𝑖, 𝐴⟩ = ⟨𝑘, 𝑗⟩)
5		2oex 8408	. . . . . . . . . . 11 ⊢ 2o ∈ V
6		opex 5412	. . . . . . . . . . 11 ⊢ ⟨𝑖, 𝐴⟩ ∈ V
7	5, 6	opth 5424	. . . . . . . . . 10 ⊢ (⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ ↔ (2o = ∅ ∧ ⟨𝑖, 𝐴⟩ = ⟨𝑘, 𝑗⟩))
8	4, 7	mtbir 323	. . . . . . . . 9 ⊢ ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩
9		goel 35543	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘∈𝑔𝑗) = ⟨∅, ⟨𝑘, 𝑗⟩⟩)
10	9	eqeq2d 2747	. . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗) ↔ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩))
11	8, 10	mtbiri 327	. . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗))
12	11	rgen2 3176	. . . . . . 7 ⊢ ∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗)
13		ralnex2 3116	. . . . . . 7 ⊢ (∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗) ↔ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗))
14	12, 13	mpbi 230	. . . . . 6 ⊢ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗)
15	14	intnan 486	. . . . 5 ⊢ ¬ (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗))
16		eqeq1 2740	. . . . . . 7 ⊢ (𝑥 = ⟨2o, ⟨𝑖, 𝐴⟩⟩ → (𝑥 = (𝑘∈𝑔𝑗) ↔ ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗)))
17	16	2rexbidv 3201	. . . . . 6 ⊢ (𝑥 = ⟨2o, ⟨𝑖, 𝐴⟩⟩ → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗)))
18		fmla0 35578	. . . . . 6 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗)}
19	17, 18	elrab2 3649	. . . . 5 ⊢ (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ (Fmla‘∅) ↔ (⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ⟨2o, ⟨𝑖, 𝐴⟩⟩ = (𝑘∈𝑔𝑗)))
20	15, 19	mtbir 323	. . . 4 ⊢ ¬ ⟨2o, ⟨𝑖, 𝐴⟩⟩ ∈ (Fmla‘∅)
21	1, 20	eqneltri 2855	. . 3 ⊢ ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅)
22		fveq2 6834	. . . 4 ⊢ (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅))
23	22	eleq2d 2822	. . 3 ⊢ (𝑁 = ∅ → (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) ↔ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅)))
24	21, 23	mtbiri 327	. 2 ⊢ (𝑁 = ∅ → ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁))
25	24	necon2ai 2961	1 ⊢ (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3440  ∅c0 4285  ⟨cop 4586  ‘cfv 6492  (class class class)co 7358  ωcom 7808  2_oc2o 8391  ∈_𝑔cgoe 35529  ∀_𝑔cgol 35531  Fmlacfmla 35533

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  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 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-map 8767  df-goel 35536  df-goal 35538  df-sat 35539  df-fmla 35541

This theorem is referenced by:  goalr  35593