Theorem goalr 35598

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 35594	. . 3 ⊢ (∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
2	1	adantl 481	. 2 ⊢ ((𝑁 ∈ ω ∧ ∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑁 ≠ ∅)
3		nnsuc 7829	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∃𝑛 ∈ ω 𝑁 = suc 𝑛)
4		suceq 6386	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
5	4	fveq2d 6839	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (Fmla‘suc 𝑥) = (Fmla‘suc ∅))
6	5	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀𝑔𝑖𝑎 ∈ (Fmla‘suc ∅)))
7	5	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc ∅)))
8	6, 7	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) → 𝑎 ∈ (Fmla‘suc ∅))))
9		suceq 6386	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
10	9	fveq2d 6839	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (Fmla‘suc 𝑥) = (Fmla‘suc 𝑦))
11	10	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑦)))
12	10	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc 𝑦)))
13	11, 12	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑦) → 𝑎 ∈ (Fmla‘suc 𝑦))))
14		suceq 6386	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
15	14	fveq2d 6839	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (Fmla‘suc 𝑥) = (Fmla‘suc suc 𝑦))
16	15	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑦)))
17	15	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc suc 𝑦)))
18	16, 17	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑦) → 𝑎 ∈ (Fmla‘suc suc 𝑦))))
19		suceq 6386	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → suc 𝑥 = suc 𝑛)
20	19	fveq2d 6839	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (Fmla‘suc 𝑥) = (Fmla‘suc 𝑛))
21	20	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛)))
22	20	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc 𝑛)))
23	21, 22	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛))))
24		peano1 7834	. . . . . . . . . 10 ⊢ ∅ ∈ ω
25		df-goal 35543	. . . . . . . . . . 11 ⊢ ∀𝑔𝑖𝑎 = ⟨2o, ⟨𝑖, 𝑎⟩⟩
26		opex 5412	. . . . . . . . . . 11 ⊢ ⟨2o, ⟨𝑖, 𝑎⟩⟩ ∈ V
27	25, 26	eqeltri 2833	. . . . . . . . . 10 ⊢ ∀𝑔𝑖𝑎 ∈ V
28		isfmlasuc 35589	. . . . . . . . . 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 693	. . . . . . . . 9 ⊢ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀𝑔𝑖𝑎 = (𝑢⊼𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢)))
30		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∀𝑔𝑖𝑎 → (𝑥 = (𝑘∈𝑔𝑗) ↔ ∀𝑔𝑖𝑎 = (𝑘∈𝑔𝑗)))
31	30	2rexbidv 3203	. . . . . . . . . . . 12 ⊢ (𝑥 = ∀𝑔𝑖𝑎 → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ∀𝑔𝑖𝑎 = (𝑘∈𝑔𝑗)))
32		fmla0 35583	. . . . . . . . . . . 12 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗)}
33	31, 32	elrab2 3638	. . . . . . . . . . 11 ⊢ (∀𝑔𝑖𝑎 ∈ (Fmla‘∅) ↔ (∀𝑔𝑖𝑎 ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ∀𝑔𝑖𝑎 = (𝑘∈𝑔𝑗)))
34	25	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → ∀𝑔𝑖𝑎 = ⟨2o, ⟨𝑖, 𝑎⟩⟩)
35		goel 35548	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘∈𝑔𝑗) = ⟨∅, ⟨𝑘, 𝑗⟩⟩)
36	34, 35	eqeq12d 2753	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (∀𝑔𝑖𝑎 = (𝑘∈𝑔𝑗) ↔ ⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩))
37		2oex 8410	. . . . . . . . . . . . . . . 16 ⊢ 2o ∈ V
38		opex 5412	. . . . . . . . . . . . . . . 16 ⊢ ⟨𝑖, 𝑎⟩ ∈ V
39	37, 38	opth 5425	. . . . . . . . . . . . . . 15 ⊢ (⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ ↔ (2o = ∅ ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑗⟩))
40		2on0 8413	. . . . . . . . . . . . . . . . 17 ⊢ 2o ≠ ∅
41		eqneqall 2944	. . . . . . . . . . . . . . . . 17 ⊢ (2o = ∅ → (2o ≠ ∅ → 𝑎 ∈ (Fmla‘suc ∅)))
42	40, 41	mpi 20	. . . . . . . . . . . . . . . 16 ⊢ (2o = ∅ → 𝑎 ∈ (Fmla‘suc ∅))
43	42	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((2o = ∅ ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑗⟩) → 𝑎 ∈ (Fmla‘suc ∅))
44	39, 43	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ → 𝑎 ∈ (Fmla‘suc ∅))
45	36, 44	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (∀𝑔𝑖𝑎 = (𝑘∈𝑔𝑗) → 𝑎 ∈ (Fmla‘suc ∅)))
46	45	rexlimdva 3139	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ω → (∃𝑗 ∈ ω ∀𝑔𝑖𝑎 = (𝑘∈𝑔𝑗) → 𝑎 ∈ (Fmla‘suc ∅)))
47	46	rexlimiv 3132	. . . . . . . . . . 11 ⊢ (∃𝑘 ∈ ω ∃𝑗 ∈ ω ∀𝑔𝑖𝑎 = (𝑘∈𝑔𝑗) → 𝑎 ∈ (Fmla‘suc ∅))
48	33, 47	simplbiim 504	. . . . . . . . . 10 ⊢ (∀𝑔𝑖𝑎 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅))
49		gonanegoal 35553	. . . . . . . . . . . . . . . 16 ⊢ (𝑢⊼𝑔𝑣) ≠ ∀𝑔𝑖𝑎
50		eqneqall 2944	. . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . 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 3139	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (Fmla‘∅) → (∃𝑣 ∈ (Fmla‘∅)∀𝑔𝑖𝑎 = (𝑢⊼𝑔𝑣) → 𝑎 ∈ (Fmla‘suc ∅)))
55		df-goal 35543	. . . . . . . . . . . . . . 15 ⊢ ∀𝑔𝑘𝑢 = ⟨2o, ⟨𝑘, 𝑢⟩⟩
56	25, 55	eqeq12i 2755	. . . . . . . . . . . . . 14 ⊢ (∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢 ↔ ⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨2o, ⟨𝑘, 𝑢⟩⟩)
57	37, 38	opth 5425	. . . . . . . . . . . . . . . . 17 ⊢ (⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨2o, ⟨𝑘, 𝑢⟩⟩ ↔ (2o = 2o ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑢⟩))
58		vex 3434	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑖 ∈ V
59		vex 3434	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑎 ∈ V
60	58, 59	opth 5425	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑢⟩ ↔ (𝑖 = 𝑘 ∧ 𝑎 = 𝑢))
61		eleq1w 2820	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑎 → (𝑢 ∈ (Fmla‘∅) ↔ 𝑎 ∈ (Fmla‘∅)))
62		fmlasssuc 35590	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∅ ∈ ω → (Fmla‘∅) ⊆ (Fmla‘suc ∅))
63	24, 62	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fmla‘∅) ⊆ (Fmla‘suc ∅)
64	63	sseli 3918	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅))
65	61, 64	biimtrdi 253	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑎 → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
66	65	eqcoms 2745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑢 → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
67	60, 66	simplbiim 504	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑢⟩ → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
68	57, 67	simplbiim 504	. . . . . . . . . . . . . . . 16 ⊢ (⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨2o, ⟨𝑘, 𝑢⟩⟩ → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
69	68	com12 32	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (Fmla‘∅) → (⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨2o, ⟨𝑘, 𝑢⟩⟩ → 𝑎 ∈ (Fmla‘suc ∅)))
70	69	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑘 ∈ ω) → (⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨2o, ⟨𝑘, 𝑢⟩⟩ → 𝑎 ∈ (Fmla‘suc ∅)))
71	56, 70	biimtrid 242	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑘 ∈ ω) → (∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢 → 𝑎 ∈ (Fmla‘suc ∅)))
72	71	rexlimdva 3139	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (Fmla‘∅) → (∃𝑘 ∈ ω ∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢 → 𝑎 ∈ (Fmla‘suc ∅)))
73	54, 72	jaod 860	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (Fmla‘∅) → ((∃𝑣 ∈ (Fmla‘∅)∀𝑔𝑖𝑎 = (𝑢⊼𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢) → 𝑎 ∈ (Fmla‘suc ∅)))
74	73	rexlimiv 3132	. . . . . . . . . 10 ⊢ (∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀𝑔𝑖𝑎 = (𝑢⊼𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢) → 𝑎 ∈ (Fmla‘suc ∅))
75	48, 74	jaoi 858	. . . . . . . . 9 ⊢ ((∀𝑔𝑖𝑎 ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀𝑔𝑖𝑎 = (𝑢⊼𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢)) → 𝑎 ∈ (Fmla‘suc ∅))
76	29, 75	sylbi 217	. . . . . . . 8 ⊢ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) → 𝑎 ∈ (Fmla‘suc ∅))
77		goalrlem 35597	. . . . . . . 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 7841	. . . . . . 7 ⊢ (𝑛 ∈ ω → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛)))
79	78	adantr 480	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛)))
80		fveq2 6835	. . . . . . . . 9 ⊢ (𝑁 = suc 𝑛 → (Fmla‘𝑁) = (Fmla‘suc 𝑛))
81	80	eleq2d 2823	. . . . . . . 8 ⊢ (𝑁 = suc 𝑛 → (∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁) ↔ ∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛)))
82	80	eleq2d 2823	. . . . . . . 8 ⊢ (𝑁 = suc 𝑛 → (𝑎 ∈ (Fmla‘𝑁) ↔ 𝑎 ∈ (Fmla‘suc 𝑛)))
83	81, 82	imbi12d 344	. . . . . . 7 ⊢ (𝑁 = suc 𝑛 → ((∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛))))
84	83	adantl 481	. . . . . 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 3131	. . . 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 451	. 2 ⊢ ((𝑁 ∈ ω ∧ ∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → (𝑁 ≠ ∅ → 𝑎 ∈ (Fmla‘𝑁)))
89	2, 88	mpd 15	1 ⊢ ((𝑁 ∈ ω ∧ ∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑎 ∈ (Fmla‘𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  ⟨cop 4574  suc csuc 6320  ‘cfv 6493  (class class class)co 7361  ωcom 7811  2_oc2o 8393  ∈_𝑔cgoe 35534  ⊼_𝑔cgna 35535  ∀_𝑔cgol 35536  Fmlacfmla 35538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-inf2 9556

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-map 8769  df-goel 35541  df-gona 35542  df-goal 35543  df-sat 35544  df-fmla 35546

This theorem is referenced by:  fmlasucdisj  35600