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Theorem goalr 34377

Description: If the "Godel-set of universal quantification" applied to a class is a Godel formula, the class is also a Godel formula. Remark: The reverse is not valid for 𝐴 being of the same height as the "Godel-set of universal quantification". (Contributed by AV, 22-Oct-2023.)

**Assertion**
Ref	Expression
goalr	⊢ ((𝑁 ∈ ω ∧ ∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑎 ∈ (Fmla‘𝑁))

Distinct variable groups: 𝑖,𝑁 𝑖,𝑎 Allowed substitution hint: 𝑁(𝑎)

Proof of Theorem goalr Dummy variables 𝑗 𝑥 𝑘 𝑢 𝑣 𝑛 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		goaln0 34373	. . 3 ⊢ (∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
2	1	adantl 483	. 2 ⊢ ((𝑁 ∈ ω ∧ ∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑁 ≠ ∅)
3		nnsuc 7870	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∃𝑛 ∈ ω 𝑁 = suc 𝑛)
4		suceq 6428	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
5	4	fveq2d 6893	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (Fmla‘suc 𝑥) = (Fmla‘suc ∅))
6	5	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀_𝑔𝑖𝑎 ∈ (Fmla‘suc ∅)))
7	5	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc ∅)))
8	6, 7	imbi12d 345	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) → 𝑎 ∈ (Fmla‘suc ∅))))
9		suceq 6428	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
10	9	fveq2d 6893	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (Fmla‘suc 𝑥) = (Fmla‘suc 𝑦))
11	10	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑦)))
12	10	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc 𝑦)))
13	11, 12	imbi12d 345	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑦) → 𝑎 ∈ (Fmla‘suc 𝑦))))
14		suceq 6428	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
15	14	fveq2d 6893	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (Fmla‘suc 𝑥) = (Fmla‘suc suc 𝑦))
16	15	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀_𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑦)))
17	15	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc suc 𝑦)))
18	16, 17	imbi12d 345	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑦) → 𝑎 ∈ (Fmla‘suc suc 𝑦))))
19		suceq 6428	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → suc 𝑥 = suc 𝑛)
20	19	fveq2d 6893	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (Fmla‘suc 𝑥) = (Fmla‘suc 𝑛))
21	20	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛)))
22	20	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc 𝑛)))
23	21, 22	imbi12d 345	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛))))
24		peano1 7876	. . . . . . . . . 10 ⊢ ∅ ∈ ω
25		df-goal 34322	. . . . . . . . . . 11 ⊢ ∀_𝑔𝑖𝑎 = ⟨2_o, ⟨𝑖, 𝑎⟩⟩
26		opex 5464	. . . . . . . . . . 11 ⊢ ⟨2_o, ⟨𝑖, 𝑎⟩⟩ ∈ V
27	25, 26	eqeltri 2830	. . . . . . . . . 10 ⊢ ∀_𝑔𝑖𝑎 ∈ V
28		isfmlasuc 34368	. . . . . . . . . 10 ⊢ ((∅ ∈ ω ∧ ∀_𝑔𝑖𝑎 ∈ V) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢))))
29	24, 27, 28	mp2an 691	. . . . . . . . 9 ⊢ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢)))
30		eqeq1 2737	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∀_𝑔𝑖𝑎 → (𝑥 = (𝑘∈_𝑔𝑗) ↔ ∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗)))
31	30	2rexbidv 3220	. . . . . . . . . . . 12 ⊢ (𝑥 = ∀_𝑔𝑖𝑎 → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈_𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗)))
32		fmla0 34362	. . . . . . . . . . . 12 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈_𝑔𝑗)}
33	31, 32	elrab2 3686	. . . . . . . . . . 11 ⊢ (∀_𝑔𝑖𝑎 ∈ (Fmla‘∅) ↔ (∀_𝑔𝑖𝑎 ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗)))
34	25	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → ∀_𝑔𝑖𝑎 = ⟨2_o, ⟨𝑖, 𝑎⟩⟩)
35		goel 34327	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘∈_𝑔𝑗) = ⟨∅, ⟨𝑘, 𝑗⟩⟩)
36	34, 35	eqeq12d 2749	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗) ↔ ⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩))
37		2oex 8474	. . . . . . . . . . . . . . . 16 ⊢ 2_o ∈ V
38		opex 5464	. . . . . . . . . . . . . . . 16 ⊢ ⟨𝑖, 𝑎⟩ ∈ V
39	37, 38	opth 5476	. . . . . . . . . . . . . . 15 ⊢ (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ ↔ (2_o = ∅ ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑗⟩))
40		2on0 8479	. . . . . . . . . . . . . . . . 17 ⊢ 2_o ≠ ∅
41		eqneqall 2952	. . . . . . . . . . . . . . . . 17 ⊢ (2_o = ∅ → (2_o ≠ ∅ → 𝑎 ∈ (Fmla‘suc ∅)))
42	40, 41	mpi 20	. . . . . . . . . . . . . . . 16 ⊢ (2_o = ∅ → 𝑎 ∈ (Fmla‘suc ∅))
43	42	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((2_o = ∅ ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑗⟩) → 𝑎 ∈ (Fmla‘suc ∅))
44	39, 43	sylbi 216	. . . . . . . . . . . . . 14 ⊢ (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ → 𝑎 ∈ (Fmla‘suc ∅))
45	36, 44	syl6bi 253	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗) → 𝑎 ∈ (Fmla‘suc ∅)))
46	45	rexlimdva 3156	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ω → (∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗) → 𝑎 ∈ (Fmla‘suc ∅)))
47	46	rexlimiv 3149	. . . . . . . . . . 11 ⊢ (∃𝑘 ∈ ω ∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗) → 𝑎 ∈ (Fmla‘suc ∅))
48	33, 47	simplbiim 506	. . . . . . . . . 10 ⊢ (∀_𝑔𝑖𝑎 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅))
49		gonanegoal 34332	. . . . . . . . . . . . . . . 16 ⊢ (𝑢⊼_𝑔𝑣) ≠ ∀_𝑔𝑖𝑎
50		eqneqall 2952	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢⊼_𝑔𝑣) = ∀_𝑔𝑖𝑎 → ((𝑢⊼_𝑔𝑣) ≠ ∀_𝑔𝑖𝑎 → 𝑎 ∈ (Fmla‘suc ∅)))
51	49, 50	mpi 20	. . . . . . . . . . . . . . 15 ⊢ ((𝑢⊼_𝑔𝑣) = ∀_𝑔𝑖𝑎 → 𝑎 ∈ (Fmla‘suc ∅))
52	51	eqcoms 2741	. . . . . . . . . . . . . 14 ⊢ (∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) → 𝑎 ∈ (Fmla‘suc ∅))
53	52	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑣 ∈ (Fmla‘∅)) → (∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) → 𝑎 ∈ (Fmla‘suc ∅)))
54	53	rexlimdva 3156	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (Fmla‘∅) → (∃𝑣 ∈ (Fmla‘∅)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) → 𝑎 ∈ (Fmla‘suc ∅)))
55		df-goal 34322	. . . . . . . . . . . . . . 15 ⊢ ∀_𝑔𝑘𝑢 = ⟨2_o, ⟨𝑘, 𝑢⟩⟩
56	25, 55	eqeq12i 2751	. . . . . . . . . . . . . 14 ⊢ (∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢 ↔ ⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑘, 𝑢⟩⟩)
57	37, 38	opth 5476	. . . . . . . . . . . . . . . . 17 ⊢ (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑘, 𝑢⟩⟩ ↔ (2_o = 2_o ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑢⟩))
58		vex 3479	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑖 ∈ V
59		vex 3479	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑎 ∈ V
60	58, 59	opth 5476	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑢⟩ ↔ (𝑖 = 𝑘 ∧ 𝑎 = 𝑢))
61		eleq1w 2817	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑎 → (𝑢 ∈ (Fmla‘∅) ↔ 𝑎 ∈ (Fmla‘∅)))
62		fmlasssuc 34369	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∅ ∈ ω → (Fmla‘∅) ⊆ (Fmla‘suc ∅))
63	24, 62	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fmla‘∅) ⊆ (Fmla‘suc ∅)
64	63	sseli 3978	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅))
65	61, 64	syl6bi 253	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑎 → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
66	65	eqcoms 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑢 → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
67	60, 66	simplbiim 506	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑢⟩ → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
68	57, 67	simplbiim 506	. . . . . . . . . . . . . . . 16 ⊢ (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑘, 𝑢⟩⟩ → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
69	68	com12 32	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (Fmla‘∅) → (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑘, 𝑢⟩⟩ → 𝑎 ∈ (Fmla‘suc ∅)))
70	69	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑘 ∈ ω) → (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑘, 𝑢⟩⟩ → 𝑎 ∈ (Fmla‘suc ∅)))
71	56, 70	biimtrid 241	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑘 ∈ ω) → (∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢 → 𝑎 ∈ (Fmla‘suc ∅)))
72	71	rexlimdva 3156	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (Fmla‘∅) → (∃𝑘 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢 → 𝑎 ∈ (Fmla‘suc ∅)))
73	54, 72	jaod 858	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (Fmla‘∅) → ((∃𝑣 ∈ (Fmla‘∅)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢) → 𝑎 ∈ (Fmla‘suc ∅)))
74	73	rexlimiv 3149	. . . . . . . . . 10 ⊢ (∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢) → 𝑎 ∈ (Fmla‘suc ∅))
75	48, 74	jaoi 856	. . . . . . . . 9 ⊢ ((∀_𝑔𝑖𝑎 ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢)) → 𝑎 ∈ (Fmla‘suc ∅))
76	29, 75	sylbi 216	. . . . . . . 8 ⊢ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) → 𝑎 ∈ (Fmla‘suc ∅))
77		goalrlem 34376	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑦) → 𝑎 ∈ (Fmla‘suc 𝑦)) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑦) → 𝑎 ∈ (Fmla‘suc suc 𝑦))))
78	8, 13, 18, 23, 76, 77	finds 7886	. . . . . . 7 ⊢ (𝑛 ∈ ω → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛)))
79	78	adantr 482	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛)))
80		fveq2 6889	. . . . . . . . 9 ⊢ (𝑁 = suc 𝑛 → (Fmla‘𝑁) = (Fmla‘suc 𝑛))
81	80	eleq2d 2820	. . . . . . . 8 ⊢ (𝑁 = suc 𝑛 → (∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) ↔ ∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛)))
82	80	eleq2d 2820	. . . . . . . 8 ⊢ (𝑁 = suc 𝑛 → (𝑎 ∈ (Fmla‘𝑁) ↔ 𝑎 ∈ (Fmla‘suc 𝑛)))
83	81, 82	imbi12d 345	. . . . . . 7 ⊢ (𝑁 = suc 𝑛 → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛))))
84	83	adantl 483	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛))))
85	79, 84	mpbird 257	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)))
86	85	rexlimiva 3148	. . . 4 ⊢ (∃𝑛 ∈ ω 𝑁 = suc 𝑛 → (∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)))
87	3, 86	syl 17	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)))
88	87	impancom 453	. 2 ⊢ ((𝑁 ∈ ω ∧ ∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → (𝑁 ≠ ∅ → 𝑎 ∈ (Fmla‘𝑁)))
89	2, 88	mpd 15	1 ⊢ ((𝑁 ∈ ω ∧ ∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑎 ∈ (Fmla‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 Vcvv 3475 ⊆ wss 3948 ∅c0 4322 ⟨cop 4634 suc csuc 6364 ‘cfv 6541 (class class class)co 7406 ωcom 7852 2_oc2o 8457 ∈_𝑔cgoe 34313 ⊼_𝑔cgna 34314 ∀_𝑔cgol 34315 Fmlacfmla 34317

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-map 8819 df-goel 34320 df-gona 34321 df-goal 34322 df-sat 34323 df-fmla 34325

This theorem is referenced by: fmlasucdisj 34379

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