gonar - Mathbox for Mario Carneiro

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Theorem gonar 33989

Description: If the "Godel-set of NAND" applied to classes is a Godel formula, the classes are also Godel formulas. Remark: The reverse is not valid for 𝐴 or 𝐵 being of the same height as the "Godel-set of NAND". (Contributed by AV, 21-Oct-2023.)

**Assertion**
Ref	Expression
gonar	⊢ ((𝑁 ∈ ω ∧ (𝑎⊼_𝑔𝑏) ∈ (Fmla‘𝑁)) → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁)))

Distinct variable group: 𝑎,𝑏 Allowed substitution hints: 𝑁(𝑎,𝑏)

Proof of Theorem gonar Dummy variables 𝑖 𝑗 𝑥 𝑢 𝑣 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gonan0 33986	. . 3 ⊢ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
2	1	adantl 482	. 2 ⊢ ((𝑁 ∈ ω ∧ (𝑎⊼_𝑔𝑏) ∈ (Fmla‘𝑁)) → 𝑁 ≠ ∅)
3		nnsuc 7820	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∃𝑥 ∈ ω 𝑁 = suc 𝑥)
4		suceq 6383	. . . . . . . . . . 11 ⊢ (𝑑 = ∅ → suc 𝑑 = suc ∅)
5	4	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑑 = ∅ → (Fmla‘suc 𝑑) = (Fmla‘suc ∅))
6	5	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑑 = ∅ → ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑑) ↔ (𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc ∅)))
7	5	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑑 = ∅ → (𝑎 ∈ (Fmla‘suc 𝑑) ↔ 𝑎 ∈ (Fmla‘suc ∅)))
8	5	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑑 = ∅ → (𝑏 ∈ (Fmla‘suc 𝑑) ↔ 𝑏 ∈ (Fmla‘suc ∅)))
9	7, 8	anbi12d 631	. . . . . . . . 9 ⊢ (𝑑 = ∅ → ((𝑎 ∈ (Fmla‘suc 𝑑) ∧ 𝑏 ∈ (Fmla‘suc 𝑑)) ↔ (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅))))
10	6, 9	imbi12d 344	. . . . . . . 8 ⊢ (𝑑 = ∅ → (((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑑) → (𝑎 ∈ (Fmla‘suc 𝑑) ∧ 𝑏 ∈ (Fmla‘suc 𝑑))) ↔ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc ∅) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅)))))
11		suceq 6383	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → suc 𝑑 = suc 𝑐)
12	11	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → (Fmla‘suc 𝑑) = (Fmla‘suc 𝑐))
13	12	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑑) ↔ (𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑐)))
14	12	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → (𝑎 ∈ (Fmla‘suc 𝑑) ↔ 𝑎 ∈ (Fmla‘suc 𝑐)))
15	12	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → (𝑏 ∈ (Fmla‘suc 𝑑) ↔ 𝑏 ∈ (Fmla‘suc 𝑐)))
16	14, 15	anbi12d 631	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → ((𝑎 ∈ (Fmla‘suc 𝑑) ∧ 𝑏 ∈ (Fmla‘suc 𝑑)) ↔ (𝑎 ∈ (Fmla‘suc 𝑐) ∧ 𝑏 ∈ (Fmla‘suc 𝑐))))
17	13, 16	imbi12d 344	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑑) → (𝑎 ∈ (Fmla‘suc 𝑑) ∧ 𝑏 ∈ (Fmla‘suc 𝑑))) ↔ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑐) → (𝑎 ∈ (Fmla‘suc 𝑐) ∧ 𝑏 ∈ (Fmla‘suc 𝑐)))))
18		suceq 6383	. . . . . . . . . . 11 ⊢ (𝑑 = suc 𝑐 → suc 𝑑 = suc suc 𝑐)
19	18	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑑 = suc 𝑐 → (Fmla‘suc 𝑑) = (Fmla‘suc suc 𝑐))
20	19	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑑 = suc 𝑐 → ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑑) ↔ (𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc suc 𝑐)))
21	19	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑑 = suc 𝑐 → (𝑎 ∈ (Fmla‘suc 𝑑) ↔ 𝑎 ∈ (Fmla‘suc suc 𝑐)))
22	19	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑑 = suc 𝑐 → (𝑏 ∈ (Fmla‘suc 𝑑) ↔ 𝑏 ∈ (Fmla‘suc suc 𝑐)))
23	21, 22	anbi12d 631	. . . . . . . . 9 ⊢ (𝑑 = suc 𝑐 → ((𝑎 ∈ (Fmla‘suc 𝑑) ∧ 𝑏 ∈ (Fmla‘suc 𝑑)) ↔ (𝑎 ∈ (Fmla‘suc suc 𝑐) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑐))))
24	20, 23	imbi12d 344	. . . . . . . 8 ⊢ (𝑑 = suc 𝑐 → (((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑑) → (𝑎 ∈ (Fmla‘suc 𝑑) ∧ 𝑏 ∈ (Fmla‘suc 𝑑))) ↔ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc suc 𝑐) → (𝑎 ∈ (Fmla‘suc suc 𝑐) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑐)))))
25		suceq 6383	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑥 → suc 𝑑 = suc 𝑥)
26	25	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑑 = 𝑥 → (Fmla‘suc 𝑑) = (Fmla‘suc 𝑥))
27	26	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑑 = 𝑥 → ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑑) ↔ (𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑥)))
28	26	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑑 = 𝑥 → (𝑎 ∈ (Fmla‘suc 𝑑) ↔ 𝑎 ∈ (Fmla‘suc 𝑥)))
29	26	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑑 = 𝑥 → (𝑏 ∈ (Fmla‘suc 𝑑) ↔ 𝑏 ∈ (Fmla‘suc 𝑥)))
30	28, 29	anbi12d 631	. . . . . . . . 9 ⊢ (𝑑 = 𝑥 → ((𝑎 ∈ (Fmla‘suc 𝑑) ∧ 𝑏 ∈ (Fmla‘suc 𝑑)) ↔ (𝑎 ∈ (Fmla‘suc 𝑥) ∧ 𝑏 ∈ (Fmla‘suc 𝑥))))
31	27, 30	imbi12d 344	. . . . . . . 8 ⊢ (𝑑 = 𝑥 → (((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑑) → (𝑎 ∈ (Fmla‘suc 𝑑) ∧ 𝑏 ∈ (Fmla‘suc 𝑑))) ↔ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑥) → (𝑎 ∈ (Fmla‘suc 𝑥) ∧ 𝑏 ∈ (Fmla‘suc 𝑥)))))
32		peano1 7825	. . . . . . . . . 10 ⊢ ∅ ∈ ω
33		ovex 7390	. . . . . . . . . 10 ⊢ (𝑎⊼_𝑔𝑏) ∈ V
34		isfmlasuc 33982	. . . . . . . . . 10 ⊢ ((∅ ∈ ω ∧ (𝑎⊼_𝑔𝑏) ∈ V) → ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc ∅) ↔ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)(𝑎⊼_𝑔𝑏) = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω (𝑎⊼_𝑔𝑏) = ∀_𝑔𝑖𝑢))))
35	32, 33, 34	mp2an 690	. . . . . . . . 9 ⊢ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc ∅) ↔ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)(𝑎⊼_𝑔𝑏) = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω (𝑎⊼_𝑔𝑏) = ∀_𝑔𝑖𝑢)))
36		eqeq1 2740	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑎⊼_𝑔𝑏) → (𝑥 = (𝑖∈_𝑔𝑗) ↔ (𝑎⊼_𝑔𝑏) = (𝑖∈_𝑔𝑗)))
37	36	2rexbidv 3213	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑎⊼_𝑔𝑏) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑎⊼_𝑔𝑏) = (𝑖∈_𝑔𝑗)))
38		fmla0 33976	. . . . . . . . . . . 12 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)}
39	37, 38	elrab2 3648	. . . . . . . . . . 11 ⊢ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘∅) ↔ ((𝑎⊼_𝑔𝑏) ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑎⊼_𝑔𝑏) = (𝑖∈_𝑔𝑗)))
40		gonafv 33944	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎⊼_𝑔𝑏) = ⟨1_o, ⟨𝑎, 𝑏⟩⟩)
41	40	el2v 3453	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎⊼_𝑔𝑏) = ⟨1_o, ⟨𝑎, 𝑏⟩⟩
42	41	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑎⊼_𝑔𝑏) = ⟨1_o, ⟨𝑎, 𝑏⟩⟩)
43		goel 33941	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈_𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
44	42, 43	eqeq12d 2752	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑎⊼_𝑔𝑏) = (𝑖∈_𝑔𝑗) ↔ ⟨1_o, ⟨𝑎, 𝑏⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
45		1oex 8422	. . . . . . . . . . . . . . . . 17 ⊢ 1_o ∈ V
46		opex 5421	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
47	45, 46	opth 5433	. . . . . . . . . . . . . . . 16 ⊢ (⟨1_o, ⟨𝑎, 𝑏⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (1_o = ∅ ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑗⟩))
48		1n0 8434	. . . . . . . . . . . . . . . . . 18 ⊢ 1_o ≠ ∅
49		eqneqall 2954	. . . . . . . . . . . . . . . . . 18 ⊢ (1_o = ∅ → (1_o ≠ ∅ → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅))))
50	48, 49	mpi 20	. . . . . . . . . . . . . . . . 17 ⊢ (1_o = ∅ → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅)))
51	50	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((1_o = ∅ ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑗⟩) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅)))
52	47, 51	sylbi 216	. . . . . . . . . . . . . . 15 ⊢ (⟨1_o, ⟨𝑎, 𝑏⟩⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅)))
53	44, 52	syl6bi 252	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑎⊼_𝑔𝑏) = (𝑖∈_𝑔𝑗) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅))))
54	53	rexlimdva 3152	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ω → (∃𝑗 ∈ ω (𝑎⊼_𝑔𝑏) = (𝑖∈_𝑔𝑗) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅))))
55	54	rexlimiv 3145	. . . . . . . . . . . 12 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑎⊼_𝑔𝑏) = (𝑖∈_𝑔𝑗) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅)))
56	55	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑎⊼_𝑔𝑏) ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑎⊼_𝑔𝑏) = (𝑖∈_𝑔𝑗)) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅)))
57	39, 56	sylbi 216	. . . . . . . . . 10 ⊢ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘∅) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅)))
58	41	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑣 ∈ (Fmla‘∅)) → (𝑎⊼_𝑔𝑏) = ⟨1_o, ⟨𝑎, 𝑏⟩⟩)
59		gonafv 33944	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑣 ∈ (Fmla‘∅)) → (𝑢⊼_𝑔𝑣) = ⟨1_o, ⟨𝑢, 𝑣⟩⟩)
60	58, 59	eqeq12d 2752	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑣 ∈ (Fmla‘∅)) → ((𝑎⊼_𝑔𝑏) = (𝑢⊼_𝑔𝑣) ↔ ⟨1_o, ⟨𝑎, 𝑏⟩⟩ = ⟨1_o, ⟨𝑢, 𝑣⟩⟩))
61	45, 46	opth 5433	. . . . . . . . . . . . . . . . 17 ⊢ (⟨1_o, ⟨𝑎, 𝑏⟩⟩ = ⟨1_o, ⟨𝑢, 𝑣⟩⟩ ↔ (1_o = 1_o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑣⟩))
62		vex 3449	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑎 ∈ V
63		vex 3449	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑏 ∈ V
64	62, 63	opth 5433	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑣⟩ ↔ (𝑎 = 𝑢 ∧ 𝑏 = 𝑣))
65		simpl 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑎 = 𝑢)
66	65	equcomd 2022	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑢 = 𝑎)
67	66	eleq1d 2822	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑢 ∈ (Fmla‘∅) ↔ 𝑎 ∈ (Fmla‘∅)))
68		simpr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
69	68	equcomd 2022	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑣 = 𝑏)
70	69	eleq1d 2822	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑣 ∈ (Fmla‘∅) ↔ 𝑏 ∈ (Fmla‘∅)))
71	67, 70	anbi12d 631	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((𝑢 ∈ (Fmla‘∅) ∧ 𝑣 ∈ (Fmla‘∅)) ↔ (𝑎 ∈ (Fmla‘∅) ∧ 𝑏 ∈ (Fmla‘∅))))
72	64, 71	sylbi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑣⟩ → ((𝑢 ∈ (Fmla‘∅) ∧ 𝑣 ∈ (Fmla‘∅)) ↔ (𝑎 ∈ (Fmla‘∅) ∧ 𝑏 ∈ (Fmla‘∅))))
73	72	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((1_o = 1_o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑣⟩) → ((𝑢 ∈ (Fmla‘∅) ∧ 𝑣 ∈ (Fmla‘∅)) ↔ (𝑎 ∈ (Fmla‘∅) ∧ 𝑏 ∈ (Fmla‘∅))))
74	61, 73	sylbi 216	. . . . . . . . . . . . . . . 16 ⊢ (⟨1_o, ⟨𝑎, 𝑏⟩⟩ = ⟨1_o, ⟨𝑢, 𝑣⟩⟩ → ((𝑢 ∈ (Fmla‘∅) ∧ 𝑣 ∈ (Fmla‘∅)) ↔ (𝑎 ∈ (Fmla‘∅) ∧ 𝑏 ∈ (Fmla‘∅))))
75		fmlasssuc 33983	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ ω → (Fmla‘∅) ⊆ (Fmla‘suc ∅))
76	32, 75	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (Fmla‘∅) ⊆ (Fmla‘suc ∅)
77	76	sseli 3940	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅))
78	76	sseli 3940	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (Fmla‘∅) → 𝑏 ∈ (Fmla‘suc ∅))
79	77, 78	anim12i 613	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ (Fmla‘∅) ∧ 𝑏 ∈ (Fmla‘∅)) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅)))
80	74, 79	syl6bi 252	. . . . . . . . . . . . . . 15 ⊢ (⟨1_o, ⟨𝑎, 𝑏⟩⟩ = ⟨1_o, ⟨𝑢, 𝑣⟩⟩ → ((𝑢 ∈ (Fmla‘∅) ∧ 𝑣 ∈ (Fmla‘∅)) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅))))
81	80	com12 32	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑣 ∈ (Fmla‘∅)) → (⟨1_o, ⟨𝑎, 𝑏⟩⟩ = ⟨1_o, ⟨𝑢, 𝑣⟩⟩ → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅))))
82	60, 81	sylbid 239	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑣 ∈ (Fmla‘∅)) → ((𝑎⊼_𝑔𝑏) = (𝑢⊼_𝑔𝑣) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅))))
83	82	rexlimdva 3152	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (Fmla‘∅) → (∃𝑣 ∈ (Fmla‘∅)(𝑎⊼_𝑔𝑏) = (𝑢⊼_𝑔𝑣) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅))))
84		gonanegoal 33946	. . . . . . . . . . . . . . 15 ⊢ (𝑎⊼_𝑔𝑏) ≠ ∀_𝑔𝑖𝑢
85		eqneqall 2954	. . . . . . . . . . . . . . 15 ⊢ ((𝑎⊼_𝑔𝑏) = ∀_𝑔𝑖𝑢 → ((𝑎⊼_𝑔𝑏) ≠ ∀_𝑔𝑖𝑢 → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅))))
86	84, 85	mpi 20	. . . . . . . . . . . . . 14 ⊢ ((𝑎⊼_𝑔𝑏) = ∀_𝑔𝑖𝑢 → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅)))
87	86	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (Fmla‘∅) ∧ 𝑖 ∈ ω) → ((𝑎⊼_𝑔𝑏) = ∀_𝑔𝑖𝑢 → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅))))
88	87	rexlimdva 3152	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (Fmla‘∅) → (∃𝑖 ∈ ω (𝑎⊼_𝑔𝑏) = ∀_𝑔𝑖𝑢 → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅))))
89	83, 88	jaod 857	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (Fmla‘∅) → ((∃𝑣 ∈ (Fmla‘∅)(𝑎⊼_𝑔𝑏) = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω (𝑎⊼_𝑔𝑏) = ∀_𝑔𝑖𝑢) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅))))
90	89	rexlimiv 3145	. . . . . . . . . 10 ⊢ (∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)(𝑎⊼_𝑔𝑏) = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω (𝑎⊼_𝑔𝑏) = ∀_𝑔𝑖𝑢) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅)))
91	57, 90	jaoi 855	. . . . . . . . 9 ⊢ (((𝑎⊼_𝑔𝑏) ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)(𝑎⊼_𝑔𝑏) = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω (𝑎⊼_𝑔𝑏) = ∀_𝑔𝑖𝑢)) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅)))
92	35, 91	sylbi 216	. . . . . . . 8 ⊢ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc ∅) → (𝑎 ∈ (Fmla‘suc ∅) ∧ 𝑏 ∈ (Fmla‘suc ∅)))
93		gonarlem 33988	. . . . . . . 8 ⊢ (𝑐 ∈ ω → (((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑐) → (𝑎 ∈ (Fmla‘suc 𝑐) ∧ 𝑏 ∈ (Fmla‘suc 𝑐))) → ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc suc 𝑐) → (𝑎 ∈ (Fmla‘suc suc 𝑐) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑐)))))
94	10, 17, 24, 31, 92, 93	finds 7835	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑥) → (𝑎 ∈ (Fmla‘suc 𝑥) ∧ 𝑏 ∈ (Fmla‘suc 𝑥))))
95	94	adantr 481	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝑁 = suc 𝑥) → ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑥) → (𝑎 ∈ (Fmla‘suc 𝑥) ∧ 𝑏 ∈ (Fmla‘suc 𝑥))))
96		fveq2 6842	. . . . . . . . 9 ⊢ (𝑁 = suc 𝑥 → (Fmla‘𝑁) = (Fmla‘suc 𝑥))
97	96	eleq2d 2823	. . . . . . . 8 ⊢ (𝑁 = suc 𝑥 → ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘𝑁) ↔ (𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑥)))
98	96	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑁 = suc 𝑥 → (𝑎 ∈ (Fmla‘𝑁) ↔ 𝑎 ∈ (Fmla‘suc 𝑥)))
99	96	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑁 = suc 𝑥 → (𝑏 ∈ (Fmla‘𝑁) ↔ 𝑏 ∈ (Fmla‘suc 𝑥)))
100	98, 99	anbi12d 631	. . . . . . . 8 ⊢ (𝑁 = suc 𝑥 → ((𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁)) ↔ (𝑎 ∈ (Fmla‘suc 𝑥) ∧ 𝑏 ∈ (Fmla‘suc 𝑥))))
101	97, 100	imbi12d 344	. . . . . . 7 ⊢ (𝑁 = suc 𝑥 → (((𝑎⊼_𝑔𝑏) ∈ (Fmla‘𝑁) → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁))) ↔ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑥) → (𝑎 ∈ (Fmla‘suc 𝑥) ∧ 𝑏 ∈ (Fmla‘suc 𝑥)))))
102	101	adantl 482	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝑁 = suc 𝑥) → (((𝑎⊼_𝑔𝑏) ∈ (Fmla‘𝑁) → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁))) ↔ ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘suc 𝑥) → (𝑎 ∈ (Fmla‘suc 𝑥) ∧ 𝑏 ∈ (Fmla‘suc 𝑥)))))
103	95, 102	mpbird 256	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑁 = suc 𝑥) → ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘𝑁) → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁))))
104	103	rexlimiva 3144	. . . 4 ⊢ (∃𝑥 ∈ ω 𝑁 = suc 𝑥 → ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘𝑁) → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁))))
105	3, 104	syl 17	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ((𝑎⊼_𝑔𝑏) ∈ (Fmla‘𝑁) → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁))))
106	105	impancom 452	. 2 ⊢ ((𝑁 ∈ ω ∧ (𝑎⊼_𝑔𝑏) ∈ (Fmla‘𝑁)) → (𝑁 ≠ ∅ → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁))))
107	2, 106	mpd 15	1 ⊢ ((𝑁 ∈ ω ∧ (𝑎⊼_𝑔𝑏) ∈ (Fmla‘𝑁)) → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 Vcvv 3445 ⊆ wss 3910 ∅c0 4282 ⟨cop 4592 suc csuc 6319 ‘cfv 6496 (class class class)co 7357 ωcom 7802 1_oc1o 8405 ∈_𝑔cgoe 33927 ⊼_𝑔cgna 33928 ∀_𝑔cgol 33929 Fmlacfmla 33931

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-map 8767 df-goel 33934 df-gona 33935 df-goal 33936 df-sat 33937 df-fmla 33939

This theorem is referenced by: fmlasucdisj 33993

W3C validator