fmlaomn0 - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fmlaomn0		Structured version Visualization version GIF version

Theorem fmlaomn0 35678

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 6852	. . . . 5 ⊢ (𝑥 = ∅ → (Fmla‘𝑥) = (Fmla‘∅))
2	1	eleq2d 2838	. . . 4 ⊢ (𝑥 = ∅ → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘∅)))
3	2	notbid 320	. . 3 ⊢ (𝑥 = ∅ → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘∅)))
4		fveq2 6852	. . . . 5 ⊢ (𝑥 = 𝑦 → (Fmla‘𝑥) = (Fmla‘𝑦))
5	4	eleq2d 2838	. . . 4 ⊢ (𝑥 = 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑦)))
6	5	notbid 320	. . 3 ⊢ (𝑥 = 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑦)))
7		fveq2 6852	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (Fmla‘𝑥) = (Fmla‘suc 𝑦))
8	7	eleq2d 2838	. . . 4 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘suc 𝑦)))
9	8	notbid 320	. . 3 ⊢ (𝑥 = suc 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘suc 𝑦)))
10		fveq2 6852	. . . . 5 ⊢ (𝑥 = 𝑁 → (Fmla‘𝑥) = (Fmla‘𝑁))
11	10	eleq2d 2838	. . . 4 ⊢ (𝑥 = 𝑁 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑁)))
12	11	notbid 320	. . 3 ⊢ (𝑥 = 𝑁 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑁)))
13		0ex 5247	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
14		opex 5421	. . . . . . . . . . . 12 ⊢ ⟨𝑖, 𝑗⟩ ∈ V
15	13, 14	pm3.2i 473	. . . . . . . . . . 11 ⊢ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V)
16	15	a1i 11	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
17		necom 3000	. . . . . . . . . . 11 ⊢ (∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ≠ ∅)
18		opnz 5431	. . . . . . . . . . 11 ⊢ (⟨∅, ⟨𝑖, 𝑗⟩⟩ ≠ ∅ ↔ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
19	17, 18	bitri 277	. . . . . . . . . 10 ⊢ (∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
20	16, 19	sylibr 236	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩)
21	20	neneqd 2952	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
22		goel 35635	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈_𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
23	22	eqeq2d 2763	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (∅ = (𝑖∈_𝑔𝑗) ↔ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
24	21, 23	mtbird 327	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = (𝑖∈_𝑔𝑗))
25	24	rgen2 3192	. . . . . 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 232	. . . . 5 ⊢ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗)
28	27	intnan 489	. . . 4 ⊢ ¬ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗))
29		fmla0 35670	. . . . . 6 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)}
30	29	eleq2i 2844	. . . . 5 ⊢ (∅ ∈ (Fmla‘∅) ↔ ∅ ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)})
31		eqeq1 2756	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 = (𝑖∈_𝑔𝑗) ↔ ∅ = (𝑖∈_𝑔𝑗)))
32	31	2rexbidv 3217	. . . . . 6 ⊢ (𝑥 = ∅ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗)))
33	32	elrab 3641	. . . . 5 ⊢ (∅ ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)} ↔ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗)))
34	30, 33	bitri 277	. . . 4 ⊢ (∅ ∈ (Fmla‘∅) ↔ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖∈_𝑔𝑗)))
35	28, 34	mtbir 325	. . 3 ⊢ ¬ ∅ ∈ (Fmla‘∅)
36		simpr 487	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∅ ∈ (Fmla‘𝑦))
37		1oex 8431	. . . . . . . . . . . . . 14 ⊢ 1_o ∈ V
38		opex 5421	. . . . . . . . . . . . . 14 ⊢ ⟨𝑢, 𝑣⟩ ∈ V
39	37, 38	opnzi 5432	. . . . . . . . . . . . 13 ⊢ ⟨1_o, ⟨𝑢, 𝑣⟩⟩ ≠ ∅
40	39	nesymi 3004	. . . . . . . . . . . 12 ⊢ ¬ ∅ = ⟨1_o, ⟨𝑢, 𝑣⟩⟩
41		gonafv 35638	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (Fmla‘𝑦) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (𝑢⊼_𝑔𝑣) = ⟨1_o, ⟨𝑢, 𝑣⟩⟩)
42	41	adantll 722	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (𝑢⊼_𝑔𝑣) = ⟨1_o, ⟨𝑢, 𝑣⟩⟩)
43	42	eqeq2d 2763	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (∅ = (𝑢⊼_𝑔𝑣) ↔ ∅ = ⟨1_o, ⟨𝑢, 𝑣⟩⟩))
44	40, 43	mtbiri 329	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → ¬ ∅ = (𝑢⊼_𝑔𝑣))
45	44	ralrimiva 3144	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → ∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣))
46		2oex 8433	. . . . . . . . . . . . . . 15 ⊢ 2_o ∈ V
47		opex 5421	. . . . . . . . . . . . . . 15 ⊢ ⟨𝑖, 𝑢⟩ ∈ V
48	46, 47	opnzi 5432	. . . . . . . . . . . . . 14 ⊢ ⟨2_o, ⟨𝑖, 𝑢⟩⟩ ≠ ∅
49	48	nesymi 3004	. . . . . . . . . . . . 13 ⊢ ¬ ∅ = ⟨2_o, ⟨𝑖, 𝑢⟩⟩
50		df-goal 35630	. . . . . . . . . . . . . 14 ⊢ ∀_𝑔𝑖𝑢 = ⟨2_o, ⟨𝑖, 𝑢⟩⟩
51	50	eqeq2i 2765	. . . . . . . . . . . . 13 ⊢ (∅ = ∀_𝑔𝑖𝑢 ↔ ∅ = ⟨2_o, ⟨𝑖, 𝑢⟩⟩)
52	49, 51	mtbir 325	. . . . . . . . . . . 12 ⊢ ¬ ∅ = ∀_𝑔𝑖𝑢
53	52	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑖 ∈ ω) → ¬ ∅ = ∀_𝑔𝑖𝑢)
54	53	ralrimiva 3144	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢)
55	45, 54	jca 518	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → (∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢))
56	55	ralrimiva 3144	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢))
57	56	adantr 483	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢))
58		ralnex 3078	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ↔ ¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣))
59		ralnex 3078	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢 ↔ ¬ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)
60	58, 59	anbi12i 636	. . . . . . . . . 10 ⊢ ((∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
61		ioran 994	. . . . . . . . . 10 ⊢ (¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
62	60, 61	bitr4i 280	. . . . . . . . 9 ⊢ ((∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢⊼_𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀_𝑔𝑖𝑢) ↔ ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
63	62	ralbii 3098	. . . . . . . 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 3078	. . . . . . . 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 277	. . . . . . 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 220	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
67		ioran 994	. . . . . 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 591	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)))
69		fmlasuc 35674	. . . . . . . 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 2838	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∅ ∈ (Fmla‘suc 𝑦) ↔ ∅ ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)})))
71		elun 4097	. . . . . . . 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 2756	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑥 = (𝑢⊼_𝑔𝑣) ↔ ∅ = (𝑢⊼_𝑔𝑣)))
73	72	rexbidv 3176	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣)))
74		eqeq1 2756	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑥 = ∀_𝑔𝑖𝑢 ↔ ∅ = ∀_𝑔𝑖𝑢))
75	74	rexbidv 3176	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))
76	73, 75	orbi12d 927	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢)))
77	76	rexbidv 3176	. . . . . . . . . 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 3629	. . . . . . . . 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 921	. . . . . . . 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 277	. . . . . . 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 289	. . . . . 6 ⊢ (𝑦 ∈ ω → (∅ ∈ (Fmla‘suc 𝑦) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀_𝑔𝑖𝑢))))
82	81	adantr 483	. . . . 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 327	. . . 4 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∅ ∈ (Fmla‘suc 𝑦))
84	83	ex 415	. . 3 ⊢ (𝑦 ∈ ω → (¬ ∅ ∈ (Fmla‘𝑦) → ¬ ∅ ∈ (Fmla‘suc 𝑦)))
85	3, 6, 9, 12, 35, 84	finds 7862	. 2 ⊢ (𝑁 ∈ ω → ¬ ∅ ∈ (Fmla‘𝑁))
86		df-nel 3052	. 2 ⊢ (∅ ∉ (Fmla‘𝑁) ↔ ¬ ∅ ∈ (Fmla‘𝑁))
87	85, 86	sylibr 236	1 ⊢ (𝑁 ∈ ω → ∅ ∉ (Fmla‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 856 = wceq 1550 ∈ wcel 2132 {cab 2730 ≠ wne 2947 ∉ wnel 3051 ∀wral 3066 ∃wrex 3076 {crab 3404 Vcvv 3444 ∪ cun 3893 ∅c0 4276 ⟨cop 4578 suc csuc 6333 ‘cfv 6506 (class class class)co 7381 ωcom 7831 1_oc1o 8414 2_oc2o 8415 ∈_𝑔cgoe 35621 ⊼_𝑔cgna 35622 ∀_𝑔cgol 35623 Fmlacfmla 35625

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-inf2 9582

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-iun 4941 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-ov 7384 df-oprab 7385 df-mpo 7386 df-om 7832 df-1st 7955 df-2nd 7956 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-2o 8422 df-map 8794 df-goel 35628 df-gona 35629 df-goal 35630 df-sat 35631 df-fmla 35633

This theorem is referenced by: fmlan0 35679 gonan0 35680

W3C validator