fmlaomn0 - Mathbox for Mario Carneiro

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Theorem fmlaomn0 34210

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 6878	. . . . 5 ⊢ (𝑥 = ∅ → (Fmla‘𝑥) = (Fmla‘∅))
2	1	eleq2d 2818	. . . 4 ⊢ (𝑥 = ∅ → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘∅)))
3	2	notbid 317	. . 3 ⊢ (𝑥 = ∅ → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘∅)))
4		fveq2 6878	. . . . 5 ⊢ (𝑥 = 𝑦 → (Fmla‘𝑥) = (Fmla‘𝑦))
5	4	eleq2d 2818	. . . 4 ⊢ (𝑥 = 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑦)))
6	5	notbid 317	. . 3 ⊢ (𝑥 = 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑦)))
7		fveq2 6878	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (Fmla‘𝑥) = (Fmla‘suc 𝑦))
8	7	eleq2d 2818	. . . 4 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘suc 𝑦)))
9	8	notbid 317	. . 3 ⊢ (𝑥 = suc 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘suc 𝑦)))
10		fveq2 6878	. . . . 5 ⊢ (𝑥 = 𝑁 → (Fmla‘𝑥) = (Fmla‘𝑁))
11	10	eleq2d 2818	. . . 4 ⊢ (𝑥 = 𝑁 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑁)))
12	11	notbid 317	. . 3 ⊢ (𝑥 = 𝑁 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑁)))
13		0ex 5300	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
14		opex 5457	. . . . . . . . . . . 12 ⊢ ⟨𝑖, 𝑗⟩ ∈ V
15	13, 14	pm3.2i 471	. . . . . . . . . . 11 ⊢ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V)
16	15	a1i 11	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
17		necom 2993	. . . . . . . . . . 11 ⊢ (∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ≠ ∅)
18		opnz 5466	. . . . . . . . . . 11 ⊢ (⟨∅, ⟨𝑖, 𝑗⟩⟩ ≠ ∅ ↔ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
19	17, 18	bitri 274	. . . . . . . . . 10 ⊢ (∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
20	16, 19	sylibr 233	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩)
21	20	neneqd 2944	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
22		goel 34167	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈_𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
23	22	eqeq2d 2742	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (∅ = (𝑖∈_𝑔𝑗) ↔ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
24	21, 23	mtbird 324	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = (𝑖∈_𝑔𝑗))
25	24	rgen2 3196	. . . . . 6 ⊢ ∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ∅ = (𝑖∈_𝑔𝑗)
26		ralnex2 3132	. . . . . 6 ⊢ (∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ∅ = (𝑖∈_𝑔𝑗) ↔ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗))
27	25, 26	mpbi 229	. . . . 5 ⊢ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗)
28	27	intnan 487	. . . 4 ⊢ ¬ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗))
29		fmla0 34202	. . . . . 6 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)}
30	29	eleq2i 2824	. . . . 5 ⊢ (∅ ∈ (Fmla‘∅) ↔ ∅ ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)})
31		eqeq1 2735	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 = (𝑖∈_𝑔𝑗) ↔ ∅ = (𝑖∈_𝑔𝑗)))
32	31	2rexbidv 3218	. . . . . 6 ⊢ (𝑥 = ∅ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗)))
33	32	elrab 3679	. . . . 5 ⊢ (∅ ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)} ↔ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗)))
34	30, 33	bitri 274	. . . 4 ⊢ (∅ ∈ (Fmla‘∅) ↔ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗)))
35	28, 34	mtbir 322	. . 3 ⊢ ¬ ∅ ∈ (Fmla‘∅)
36		simpr 485	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∅ ∈ (Fmla‘𝑦))
37		1oex 8458	. . . . . . . . . . . . . 14 ⊢ 1_o ∈ V
38		opex 5457	. . . . . . . . . . . . . 14 ⊢ ⟨𝑢, 𝑣⟩ ∈ V
39	37, 38	opnzi 5467	. . . . . . . . . . . . 13 ⊢ ⟨1_o, ⟨𝑢, 𝑣⟩⟩ ≠ ∅
40	39	nesymi 2997	. . . . . . . . . . . 12 ⊢ ¬ ∅ = ⟨1_o, ⟨𝑢, 𝑣⟩⟩
41		gonafv 34170	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (Fmla‘𝑦) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (𝑢⊼_𝑔𝑣) = ⟨1_o, ⟨𝑢, 𝑣⟩⟩)
42	41	adantll 712	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (𝑢⊼_𝑔𝑣) = ⟨1_o, ⟨𝑢, 𝑣⟩⟩)
43	42	eqeq2d 2742	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (∅ = (𝑢⊼_𝑔𝑣) ↔ ∅ = ⟨1_o, ⟨𝑢, 𝑣⟩⟩))
44	40, 43	mtbiri 326	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → ¬ ∅ = (𝑢⊼_𝑔𝑣))
45	44	ralrimiva 3145	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → ∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣))
46		2oex 8459	. . . . . . . . . . . . . . 15 ⊢ 2_o ∈ V
47		opex 5457	. . . . . . . . . . . . . . 15 ⊢ ⟨𝑖, 𝑢⟩ ∈ V
48	46, 47	opnzi 5467	. . . . . . . . . . . . . 14 ⊢ ⟨2_o, ⟨𝑖, 𝑢⟩⟩ ≠ ∅
49	48	nesymi 2997	. . . . . . . . . . . . 13 ⊢ ¬ ∅ = ⟨2_o, ⟨𝑖, 𝑢⟩⟩
50		df-goal 34162	. . . . . . . . . . . . . 14 ⊢ ∀_𝑔𝑖𝑢 = ⟨2_o, ⟨𝑖, 𝑢⟩⟩
51	50	eqeq2i 2744	. . . . . . . . . . . . 13 ⊢ (∅ = ∀_𝑔𝑖𝑢 ↔ ∅ = ⟨2_o, ⟨𝑖, 𝑢⟩⟩)
52	49, 51	mtbir 322	. . . . . . . . . . . 12 ⊢ ¬ ∅ = ∀_𝑔𝑖𝑢
53	52	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑖 ∈ ω) → ¬ ∅ = ∀_𝑔𝑖𝑢)
54	53	ralrimiva 3145	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢)
55	45, 54	jca 512	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → (∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢))
56	55	ralrimiva 3145	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢))
57	56	adantr 481	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢))
58		ralnex 3071	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ↔ ¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣))
59		ralnex 3071	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢 ↔ ¬ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)
60	58, 59	anbi12i 627	. . . . . . . . . 10 ⊢ ((∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
61		ioran 982	. . . . . . . . . 10 ⊢ (¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
62	60, 61	bitr4i 277	. . . . . . . . 9 ⊢ ((∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢) ↔ ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
63	62	ralbii 3092	. . . . . . . 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 3071	. . . . . . . 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 274	. . . . . . 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 217	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
67		ioran 982	. . . . . 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 583	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)))
69		fmlasuc 34206	. . . . . . . 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 2818	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∅ ∈ (Fmla‘suc 𝑦) ↔ ∅ ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)})))
71		elun 4144	. . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑥 = (𝑢⊼_𝑔𝑣) ↔ ∅ = (𝑢⊼_𝑔𝑣)))
73	72	rexbidv 3177	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣)))
74		eqeq1 2735	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑥 = ∀_𝑔𝑖𝑢 ↔ ∅ = ∀_𝑔𝑖𝑢))
75	74	rexbidv 3177	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
76	73, 75	orbi12d 917	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)))
77	76	rexbidv 3177	. . . . . . . . . 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 3664	. . . . . . . . 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 911	. . . . . . . 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 274	. . . . . . 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 286	. . . . . 6 ⊢ (𝑦 ∈ ω → (∅ ∈ (Fmla‘suc 𝑦) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))))
82	81	adantr 481	. . . . 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 324	. . . 4 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∅ ∈ (Fmla‘suc 𝑦))
84	83	ex 413	. . 3 ⊢ (𝑦 ∈ ω → (¬ ∅ ∈ (Fmla‘𝑦) → ¬ ∅ ∈ (Fmla‘suc 𝑦)))
85	3, 6, 9, 12, 35, 84	finds 7871	. 2 ⊢ (𝑁 ∈ ω → ¬ ∅ ∈ (Fmla‘𝑁))
86		df-nel 3046	. 2 ⊢ (∅ ∉ (Fmla‘𝑁) ↔ ¬ ∅ ∈ (Fmla‘𝑁))
87	85, 86	sylibr 233	1 ⊢ (𝑁 ∈ ω → ∅ ∉ (Fmla‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 {cab 2708 ≠ wne 2939 ∉ wnel 3045 ∀wral 3060 ∃wrex 3069 {crab 3431 Vcvv 3473 ∪ cun 3942 ∅c0 4318 ⟨cop 4628 suc csuc 6355 ‘cfv 6532 (class class class)co 7393 ωcom 7838 1_oc1o 8441 2_oc2o 8442 ∈_𝑔cgoe 34153 ⊼_𝑔cgna 34154 ∀_𝑔cgol 34155 Fmlacfmla 34157

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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-inf2 9618

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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-2o 8449 df-map 8805 df-goel 34160 df-gona 34161 df-goal 34162 df-sat 34163 df-fmla 34165

This theorem is referenced by: fmlan0 34211 gonan0 34212

W3C validator