fmlaomn0 - Mathbox for Mario Carneiro

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Theorem fmlaomn0 35890

Description: The empty set is not a Godel formula of any height. (Contributed by AV, 21-Oct-2023.)

**Assertion**
Ref	Expression
fmlaomn0	⊢ (𝑁 ∈ ω → ∅ ∉ (Fmla‘𝑁))

Proof of Theorem fmlaomn0 Dummy variables 𝑥 𝑖 𝑗 𝑢 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . 5 ⊢ (𝑥 = ∅ → (Fmla‘𝑥) = (Fmla‘∅))
2	1	eleq2d 2848	. . . 4 ⊢ (𝑥 = ∅ → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘∅)))
3	2	notbid 321	. . 3 ⊢ (𝑥 = ∅ → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘∅)))
4		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝑦 → (Fmla‘𝑥) = (Fmla‘𝑦))
5	4	eleq2d 2848	. . . 4 ⊢ (𝑥 = 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑦)))
6	5	notbid 321	. . 3 ⊢ (𝑥 = 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑦)))
7		fveq2 6881	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (Fmla‘𝑥) = (Fmla‘suc 𝑦))
8	7	eleq2d 2848	. . . 4 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘suc 𝑦)))
9	8	notbid 321	. . 3 ⊢ (𝑥 = suc 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘suc 𝑦)))
10		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝑁 → (Fmla‘𝑥) = (Fmla‘𝑁))
11	10	eleq2d 2848	. . . 4 ⊢ (𝑥 = 𝑁 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑁)))
12	11	notbid 321	. . 3 ⊢ (𝑥 = 𝑁 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑁)))
13		0ex 5269	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
14		opex 5444	. . . . . . . . . . . 12 ⊢ ⟨𝑖, 𝑗⟩ ∈ V
15	13, 14	pm3.2i 475	. . . . . . . . . . 11 ⊢ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V)
16	15	a1i 11	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
17		necom 3010	. . . . . . . . . . 11 ⊢ (∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ≠ ∅)
18		opnz 5454	. . . . . . . . . . 11 ⊢ (⟨∅, ⟨𝑖, 𝑗⟩⟩ ≠ ∅ ↔ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
19	17, 18	bitri 278	. . . . . . . . . 10 ⊢ (∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
20	16, 19	sylibr 237	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩)
21	20	neneqd 2962	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
22		goel 35847	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈_𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
23	22	eqeq2d 2773	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (∅ = (𝑖∈_𝑔𝑗) ↔ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
24	21, 23	mtbird 328	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = (𝑖∈_𝑔𝑗))
25	24	rgen2 3204	. . . . . 6 ⊢ ∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ∅ = (𝑖∈_𝑔𝑗)
26		ralnex2 3144	. . . . . 6 ⊢ (∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ∅ = (𝑖∈_𝑔𝑗) ↔ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗))
27	25, 26	mpbi 233	. . . . 5 ⊢ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗)
28	27	intnan 491	. . . 4 ⊢ ¬ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗))
29		fmla0 35882	. . . . . 6 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)}
30	29	eleq2i 2854	. . . . 5 ⊢ (∅ ∈ (Fmla‘∅) ↔ ∅ ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)})
31		eqeq1 2766	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 = (𝑖∈_𝑔𝑗) ↔ ∅ = (𝑖∈_𝑔𝑗)))
32	31	2rexbidv 3229	. . . . . 6 ⊢ (𝑥 = ∅ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗)))
33	32	elrab 3649	. . . . 5 ⊢ (∅ ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)} ↔ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗)))
34	30, 33	bitri 278	. . . 4 ⊢ (∅ ∈ (Fmla‘∅) ↔ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗)))
35	28, 34	mtbir 326	. . 3 ⊢ ¬ ∅ ∈ (Fmla‘∅)
36		simpr 489	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∅ ∈ (Fmla‘𝑦))
37		1oex 8461	. . . . . . . . . . . . . 14 ⊢ 1_o ∈ V
38		opex 5444	. . . . . . . . . . . . . 14 ⊢ ⟨𝑢, 𝑣⟩ ∈ V
39	37, 38	opnzi 5455	. . . . . . . . . . . . 13 ⊢ ⟨1_o, ⟨𝑢, 𝑣⟩⟩ ≠ ∅
40	39	nesymi 3014	. . . . . . . . . . . 12 ⊢ ¬ ∅ = ⟨1_o, ⟨𝑢, 𝑣⟩⟩
41		gonafv 35850	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (Fmla‘𝑦) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (𝑢⊼_𝑔𝑣) = ⟨1_o, ⟨𝑢, 𝑣⟩⟩)
42	41	adantll 726	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (𝑢⊼_𝑔𝑣) = ⟨1_o, ⟨𝑢, 𝑣⟩⟩)
43	42	eqeq2d 2773	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (∅ = (𝑢⊼_𝑔𝑣) ↔ ∅ = ⟨1_o, ⟨𝑢, 𝑣⟩⟩))
44	40, 43	mtbiri 330	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → ¬ ∅ = (𝑢⊼_𝑔𝑣))
45	44	ralrimiva 3156	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → ∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣))
46		2oex 8463	. . . . . . . . . . . . . . 15 ⊢ 2_o ∈ V
47		opex 5444	. . . . . . . . . . . . . . 15 ⊢ ⟨𝑖, 𝑢⟩ ∈ V
48	46, 47	opnzi 5455	. . . . . . . . . . . . . 14 ⊢ ⟨2_o, ⟨𝑖, 𝑢⟩⟩ ≠ ∅
49	48	nesymi 3014	. . . . . . . . . . . . 13 ⊢ ¬ ∅ = ⟨2_o, ⟨𝑖, 𝑢⟩⟩
50		df-goal 35842	. . . . . . . . . . . . . 14 ⊢ ∀_𝑔𝑖𝑢 = ⟨2_o, ⟨𝑖, 𝑢⟩⟩
51	50	eqeq2i 2775	. . . . . . . . . . . . 13 ⊢ (∅ = ∀_𝑔𝑖𝑢 ↔ ∅ = ⟨2_o, ⟨𝑖, 𝑢⟩⟩)
52	49, 51	mtbir 326	. . . . . . . . . . . 12 ⊢ ¬ ∅ = ∀_𝑔𝑖𝑢
53	52	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑖 ∈ ω) → ¬ ∅ = ∀_𝑔𝑖𝑢)
54	53	ralrimiva 3156	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢)
55	45, 54	jca 520	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → (∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢))
56	55	ralrimiva 3156	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢))
57	56	adantr 485	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢))
58		ralnex 3090	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ↔ ¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣))
59		ralnex 3090	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢 ↔ ¬ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)
60	58, 59	anbi12i 639	. . . . . . . . . 10 ⊢ ((∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
61		ioran 998	. . . . . . . . . 10 ⊢ (¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
62	60, 61	bitr4i 281	. . . . . . . . 9 ⊢ ((∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢) ↔ ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
63	62	ralbii 3110	. . . . . . . 8 ⊢ (∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢) ↔ ∀𝑢 ∈ (Fmla‘𝑦) ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
64		ralnex 3090	. . . . . . . 8 ⊢ (∀𝑢 ∈ (Fmla‘𝑦) ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢) ↔ ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
65	63, 64	bitri 278	. . . . . . 7 ⊢ (∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢) ↔ ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
66	57, 65	sylib 221	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
67		ioran 998	. . . . . 6 ⊢ (¬ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)) ↔ (¬ ∅ ∈ (Fmla‘𝑦) ∧ ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)))
68	36, 66, 67	sylanbrc 594	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)))
69		fmlasuc 35886	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (Fmla‘suc 𝑦) = ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)}))
70	69	eleq2d 2848	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∅ ∈ (Fmla‘suc 𝑦) ↔ ∅ ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)})))
71		elun 4106	. . . . . . . 8 ⊢ (∅ ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)}) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∅ ∈ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)}))
72		eqeq1 2766	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑥 = (𝑢⊼_𝑔𝑣) ↔ ∅ = (𝑢⊼_𝑔𝑣)))
73	72	rexbidv 3188	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣)))
74		eqeq1 2766	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑥 = ∀_𝑔𝑖𝑢 ↔ ∅ = ∀_𝑔𝑖𝑢))
75	74	rexbidv 3188	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
76	73, 75	orbi12d 931	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)))
77	76	rexbidv 3188	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢) ↔ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)))
78	13, 77	elab 3637	. . . . . . . . 9 ⊢ (∅ ∈ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)} ↔ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
79	78	orbi2i 925	. . . . . . . 8 ⊢ ((∅ ∈ (Fmla‘𝑦) ∨ ∅ ∈ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)}) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)))
80	71, 79	bitri 278	. . . . . . 7 ⊢ (∅ ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)}) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)))
81	70, 80	bitrdi 290	. . . . . 6 ⊢ (𝑦 ∈ ω → (∅ ∈ (Fmla‘suc 𝑦) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))))
82	81	adantr 485	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → (∅ ∈ (Fmla‘suc 𝑦) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))))
83	68, 82	mtbird 328	. . . 4 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∅ ∈ (Fmla‘suc 𝑦))
84	83	ex 417	. . 3 ⊢ (𝑦 ∈ ω → (¬ ∅ ∈ (Fmla‘𝑦) → ¬ ∅ ∈ (Fmla‘suc 𝑦)))
85	3, 6, 9, 12, 35, 84	finds 7891	. 2 ⊢ (𝑁 ∈ ω → ¬ ∅ ∈ (Fmla‘𝑁))
86		df-nel 3064	. 2 ⊢ (∅ ∉ (Fmla‘𝑁) ↔ ¬ ∅ ∈ (Fmla‘𝑁))
87	85, 86	sylibr 237	1 ⊢ (𝑁 ∈ ω → ∅ ∉ (Fmla‘𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1569 ∈ wcel 2142 {cab 2740 ≠ wne 2957 ∉ wnel 3063 ∀wral 3078 ∃wrex 3088 {crab 3415 Vcvv 3454 ∪ cun 3902 ∅c0 4285 ⟨cop 4594 suc csuc 6362 ‘cfv 6536 (class class class)co 7412 ωcom 7860 1_oc1o 8444 2_oc2o 8445 ∈_𝑔cgoe 35833 ⊼_𝑔cgna 35834 ∀_𝑔cgol 35835 Fmlacfmla 35837

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-map 8824 df-goel 35840 df-gona 35841 df-goal 35842 df-sat 35843 df-fmla 35845

This theorem is used by: fmlan0 35891 gonan0 35892

W3C validator