goalr - Mathbox for Mario Carneiro

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Theorem goalr 35926

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 35922	. . 3 ⊢ (∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
2	1	adantl 487	. 2 ⊢ ((𝑁 ∈ ω ∧ ∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑁 ≠ ∅)
3		nnsuc 7886	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∃𝑛 ∈ ω 𝑁 = suc 𝑛)
4		suceq 6433	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
5	4	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (Fmla‘suc 𝑥) = (Fmla‘suc ∅))
6	5	eleq2d 2851	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀_𝑔𝑖𝑎 ∈ (Fmla‘suc ∅)))
7	5	eleq2d 2851	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc ∅)))
8	6, 7	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) → 𝑎 ∈ (Fmla‘suc ∅))))
9		suceq 6433	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
10	9	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (Fmla‘suc 𝑥) = (Fmla‘suc 𝑦))
11	10	eleq2d 2851	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑦)))
12	10	eleq2d 2851	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc 𝑦)))
13	11, 12	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑦) → 𝑎 ∈ (Fmla‘suc 𝑦))))
14		suceq 6433	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
15	14	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (Fmla‘suc 𝑥) = (Fmla‘suc suc 𝑦))
16	15	eleq2d 2851	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀_𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑦)))
17	15	eleq2d 2851	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc suc 𝑦)))
18	16, 17	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑦) → 𝑎 ∈ (Fmla‘suc suc 𝑦))))
19		suceq 6433	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → suc 𝑥 = suc 𝑛)
20	19	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (Fmla‘suc 𝑥) = (Fmla‘suc 𝑛))
21	20	eleq2d 2851	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛)))
22	20	eleq2d 2851	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc 𝑛)))
23	21, 22	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛))))
24		peano1 7891	. . . . . . . . . 10 ⊢ ∅ ∈ ω
25		df-goal 35871	. . . . . . . . . . 11 ⊢ ∀_𝑔𝑖𝑎 = ⟨2_o, ⟨𝑖, 𝑎⟩⟩
26		opex 5447	. . . . . . . . . . 11 ⊢ ⟨2_o, ⟨𝑖, 𝑎⟩⟩ ∈ V
27	25, 26	eqeltri 2861	. . . . . . . . . 10 ⊢ ∀_𝑔𝑖𝑎 ∈ V
28		isfmlasuc 35917	. . . . . . . . . 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 705	. . . . . . . . 9 ⊢ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢)))
30		eqeq1 2769	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∀_𝑔𝑖𝑎 → (𝑥 = (𝑘∈_𝑔𝑗) ↔ ∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗)))
31	30	2rexbidv 3232	. . . . . . . . . . . 12 ⊢ (𝑥 = ∀_𝑔𝑖𝑎 → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈_𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗)))
32		fmla0 35911	. . . . . . . . . . . 12 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈_𝑔𝑗)}
33	31, 32	elrab2 3656	. . . . . . . . . . 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 35876	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘∈_𝑔𝑗) = ⟨∅, ⟨𝑘, 𝑗⟩⟩)
36	34, 35	eqeq12d 2781	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗) ↔ ⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩))
37		2oex 8471	. . . . . . . . . . . . . . . 16 ⊢ 2_o ∈ V
38		opex 5447	. . . . . . . . . . . . . . . 16 ⊢ ⟨𝑖, 𝑎⟩ ∈ V
39	37, 38	opth 5460	. . . . . . . . . . . . . . 15 ⊢ (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ ↔ (2_o = ∅ ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑗⟩))
40		2on0 8474	. . . . . . . . . . . . . . . . 17 ⊢ 2_o ≠ ∅
41		eqneqall 2971	. . . . . . . . . . . . . . . . 17 ⊢ (2_o = ∅ → (2_o ≠ ∅ → 𝑎 ∈ (Fmla‘suc ∅)))
42	40, 41	mpi 21	. . . . . . . . . . . . . . . 16 ⊢ (2_o = ∅ → 𝑎 ∈ (Fmla‘suc ∅))
43	42	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((2_o = ∅ ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑗⟩) → 𝑎 ∈ (Fmla‘suc ∅))
44	39, 43	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ → 𝑎 ∈ (Fmla‘suc ∅))
45	36, 44	biimtrdi 256	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗) → 𝑎 ∈ (Fmla‘suc ∅)))
46	45	rexlimdva 3168	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ω → (∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗) → 𝑎 ∈ (Fmla‘suc ∅)))
47	46	rexlimiv 3161	. . . . . . . . . . 11 ⊢ (∃𝑘 ∈ ω ∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = (𝑘∈_𝑔𝑗) → 𝑎 ∈ (Fmla‘suc ∅))
48	33, 47	simplbiim 514	. . . . . . . . . 10 ⊢ (∀_𝑔𝑖𝑎 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅))
49		gonanegoal 35881	. . . . . . . . . . . . . . . 16 ⊢ (𝑢⊼_𝑔𝑣) ≠ ∀_𝑔𝑖𝑎
50		eqneqall 2971	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢⊼_𝑔𝑣) = ∀_𝑔𝑖𝑎 → ((𝑢⊼_𝑔𝑣) ≠ ∀_𝑔𝑖𝑎 → 𝑎 ∈ (Fmla‘suc ∅)))
51	49, 50	mpi 21	. . . . . . . . . . . . . . 15 ⊢ ((𝑢⊼_𝑔𝑣) = ∀_𝑔𝑖𝑎 → 𝑎 ∈ (Fmla‘suc ∅))
52	51	eqcoms 2773	. . . . . . . . . . . . . 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 3168	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (Fmla‘∅) → (∃𝑣 ∈ (Fmla‘∅)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) → 𝑎 ∈ (Fmla‘suc ∅)))
55		df-goal 35871	. . . . . . . . . . . . . . 15 ⊢ ∀_𝑔𝑘𝑢 = ⟨2_o, ⟨𝑘, 𝑢⟩⟩
56	25, 55	eqeq12i 2783	. . . . . . . . . . . . . 14 ⊢ (∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢 ↔ ⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑘, 𝑢⟩⟩)
57	37, 38	opth 5460	. . . . . . . . . . . . . . . . 17 ⊢ (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑘, 𝑢⟩⟩ ↔ (2_o = 2_o ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑢⟩))
58		vex 3461	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑖 ∈ V
59		vex 3461	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑎 ∈ V
60	58, 59	opth 5460	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑢⟩ ↔ (𝑖 = 𝑘 ∧ 𝑎 = 𝑢))
61		eleq1w 2848	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑎 → (𝑢 ∈ (Fmla‘∅) ↔ 𝑎 ∈ (Fmla‘∅)))
62		fmlasssuc 35918	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∅ ∈ ω → (Fmla‘∅) ⊆ (Fmla‘suc ∅))
63	24, 62	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fmla‘∅) ⊆ (Fmla‘suc ∅)
64	63	sseli 3934	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅))
65	61, 64	biimtrdi 256	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑎 → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
66	65	eqcoms 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑢 → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
67	60, 66	simplbiim 514	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑢⟩ → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
68	57, 67	simplbiim 514	. . . . . . . . . . . . . . . 16 ⊢ (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑘, 𝑢⟩⟩ → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
69	68	com12 33	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (Fmla‘∅) → (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑘, 𝑢⟩⟩ → 𝑎 ∈ (Fmla‘suc ∅)))
70	69	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑘 ∈ ω) → (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑘, 𝑢⟩⟩ → 𝑎 ∈ (Fmla‘suc ∅)))
71	56, 70	biimtrid 245	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑘 ∈ ω) → (∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢 → 𝑎 ∈ (Fmla‘suc ∅)))
72	71	rexlimdva 3168	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (Fmla‘∅) → (∃𝑘 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢 → 𝑎 ∈ (Fmla‘suc ∅)))
73	54, 72	jaod 873	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (Fmla‘∅) → ((∃𝑣 ∈ (Fmla‘∅)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢) → 𝑎 ∈ (Fmla‘suc ∅)))
74	73	rexlimiv 3161	. . . . . . . . . 10 ⊢ (∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢) → 𝑎 ∈ (Fmla‘suc ∅))
75	48, 74	jaoi 871	. . . . . . . . 9 ⊢ ((∀_𝑔𝑖𝑎 ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑘𝑢)) → 𝑎 ∈ (Fmla‘suc ∅))
76	29, 75	sylbi 220	. . . . . . . 8 ⊢ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) → 𝑎 ∈ (Fmla‘suc ∅))
77		goalrlem 35925	. . . . . . . 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 7899	. . . . . . 7 ⊢ (𝑛 ∈ ω → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛)))
79	78	adantr 486	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛)))
80		fveq2 6885	. . . . . . . . 9 ⊢ (𝑁 = suc 𝑛 → (Fmla‘𝑁) = (Fmla‘suc 𝑛))
81	80	eleq2d 2851	. . . . . . . 8 ⊢ (𝑁 = suc 𝑛 → (∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) ↔ ∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛)))
82	80	eleq2d 2851	. . . . . . . 8 ⊢ (𝑁 = suc 𝑛 → (𝑎 ∈ (Fmla‘𝑁) ↔ 𝑎 ∈ (Fmla‘suc 𝑛)))
83	81, 82	imbi12d 347	. . . . . . 7 ⊢ (𝑁 = suc 𝑛 → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛))))
84	83	adantl 487	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛))))
85	79, 84	mpbird 260	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)))
86	85	rexlimiva 3160	. . . 4 ⊢ (∃𝑛 ∈ ω 𝑁 = suc 𝑛 → (∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)))
87	3, 86	syl 18	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)))
88	87	impancom 457	. 2 ⊢ ((𝑁 ∈ ω ∧ ∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → (𝑁 ≠ ∅ → 𝑎 ∈ (Fmla‘𝑁)))
89	2, 88	mpd 16	1 ⊢ ((𝑁 ∈ ω ∧ ∀_𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑎 ∈ (Fmla‘𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∃wrex 3091 Vcvv 3457 ⊆ wss 3906 ∅c0 4286 ⟨cop 4597 suc csuc 6366 ‘cfv 6540 (class class class)co 7419 ωcom 7868 2_oc2o 8453 ∈_𝑔cgoe 35862 ⊼_𝑔cgna 35863 ∀_𝑔cgol 35864 Fmlacfmla 35866

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 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 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-map 8832 df-goel 35869 df-gona 35870 df-goal 35871 df-sat 35872 df-fmla 35874

This theorem is used by: fmlasucdisj 35928

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