fmlafvel - Mathbox for Mario Carneiro

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Theorem fmlafvel 35972

Description: A class is a valid Godel formula of height 𝑁 iff it is the first component of a member of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 19-Sep-2023.)

**Assertion**
Ref	Expression
fmlafvel	⊢ (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))

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

Step	Hyp	Ref	Expression
1		fveq2 6882	. . . . . . 7 ⊢ (𝑥 = ∅ → (Fmla‘𝑥) = (Fmla‘∅))
2	1	eleq2d 2848	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹 ∈ (Fmla‘𝑥) ↔ 𝐹 ∈ (Fmla‘∅)))
3		fveq2 6882	. . . . . . 7 ⊢ (𝑥 = ∅ → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘∅))
4	3	eleq2d 2848	. . . . . 6 ⊢ (𝑥 = ∅ → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅)))
5	2, 4	bibi12d 348	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥)) ↔ (𝐹 ∈ (Fmla‘∅) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅))))
6	5	imbi2d 343	. . . 4 ⊢ (𝑥 = ∅ → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥))) ↔ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘∅) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅)))))
7		fveq2 6882	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (Fmla‘𝑥) = (Fmla‘𝑦))
8	7	eleq2d 2848	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ∈ (Fmla‘𝑥) ↔ 𝐹 ∈ (Fmla‘𝑦)))
9		fveq2 6882	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑦))
10	9	eleq2d 2848	. . . . . 6 ⊢ (𝑥 = 𝑦 → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))
11	8, 10	bibi12d 348	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥)) ↔ (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦))))
12	11	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥))) ↔ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))))
13		fveq2 6882	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (Fmla‘𝑥) = (Fmla‘suc 𝑦))
14	13	eleq2d 2848	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐹 ∈ (Fmla‘𝑥) ↔ 𝐹 ∈ (Fmla‘suc 𝑦)))
15		fveq2 6882	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘suc 𝑦))
16	15	eleq2d 2848	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦)))
17	14, 16	bibi12d 348	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥)) ↔ (𝐹 ∈ (Fmla‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦))))
18	17	imbi2d 343	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥))) ↔ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦)))))
19		fveq2 6882	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (Fmla‘𝑥) = (Fmla‘𝑁))
20	19	eleq2d 2848	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐹 ∈ (Fmla‘𝑥) ↔ 𝐹 ∈ (Fmla‘𝑁)))
21		fveq2 6882	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑁))
22	21	eleq2d 2848	. . . . . 6 ⊢ (𝑥 = 𝑁 → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
23	20, 22	bibi12d 348	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥)) ↔ (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
24	23	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥))) ↔ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))))
25		eqeq1 2766	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥 = (𝑖∈_𝑔𝑗) ↔ 𝐹 = (𝑖∈_𝑔𝑗)))
26	25	2rexbidv 3229	. . . . . . 7 ⊢ (𝑥 = 𝐹 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗)))
27	26	elrab 3648	. . . . . 6 ⊢ (𝐹 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)} ↔ (𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗)))
28		eqidd 2763	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗)) → ∅ = ∅)
29		simpr 490	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗))
30	28, 29	jca 521	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗)) → (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗)))
31		simpr 490	. . . . . . . . 9 ⊢ ((∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗))
32	31	anim2i 629	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗))) → (𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗)))
33	32	ex 418	. . . . . . 7 ⊢ (𝐹 ∈ V → ((∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗)) → (𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗))))
34	30, 33	impbid2 229	. . . . . 6 ⊢ (𝐹 ∈ V → ((𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗)) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗))))
35	27, 34	bitrid 286	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)} ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗))))
36		fmla0 35969	. . . . . . 7 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)}
37	36	eleq2i 2854	. . . . . 6 ⊢ (𝐹 ∈ (Fmla‘∅) ↔ 𝐹 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)})
38	37	a1i 11	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘∅) ↔ 𝐹 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)}))
39		satf00 35961	. . . . . . . 8 ⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗))}
40	39	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ V → ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗))})
41	40	eleq2d 2848	. . . . . 6 ⊢ (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅) ↔ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗))}))
42		0ex 5268	. . . . . . 7 ⊢ ∅ ∈ V
43		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑦 = ∅ ↔ ∅ = ∅))
44	43, 26	bi2anan9r 651	. . . . . . . 8 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = ∅) → ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗))))
45	44	opelopabga 5515	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ ∅ ∈ V) → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗))} ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗))))
46	42, 45	mpan2 704	. . . . . 6 ⊢ (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗))} ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗))))
47	41, 46	bitrd 282	. . . . 5 ⊢ (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈_𝑔𝑗))))
48	35, 38, 47	3bitr4d 314	. . . 4 ⊢ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘∅) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅)))
49		eqid 2762	. . . . . . . . . . . 12 ⊢ ∅ = ∅
50	49	biantrur 540	. . . . . . . . . . 11 ⊢ (∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢)) ↔ (∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢))))
51	50	bicomi 227	. . . . . . . . . 10 ⊢ ((∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢))) ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢)))
52	51	a1i 11	. . . . . . . . 9 ⊢ (𝐹 ∈ V → ((∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢))) ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢))))
53		eqeq1 2766	. . . . . . . . . . . 12 ⊢ (𝑧 = ∅ → (𝑧 = ∅ ↔ ∅ = ∅))
54		eqeq1 2766	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐹 → (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ↔ 𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣))))
55	54	rexbidv 3188	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐹 → (∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ↔ ∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣))))
56		eqeq1 2766	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐹 → (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ↔ 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢)))
57	56	rexbidv 3188	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐹 → (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ↔ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢)))
58	55, 57	orbi12d 932	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐹 → ((∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)) ↔ (∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢))))
59	58	rexbidv 3188	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐹 → (∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)) ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢))))
60	53, 59	bi2anan9r 651	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐹 ∧ 𝑧 = ∅) → ((𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))) ↔ (∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢)))))
61	60	opelopabga 5515	. . . . . . . . . 10 ⊢ ((𝐹 ∈ V ∧ ∅ ∈ V) → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))} ↔ (∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢)))))
62	42, 61	mpan2 704	. . . . . . . . 9 ⊢ (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))} ↔ (∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢)))))
63	59	elabg 3633	. . . . . . . . 9 ⊢ (𝐹 ∈ V → (𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))} ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀_𝑔𝑖(1^st ‘𝑢))))
64	52, 62, 63	3bitr4d 314	. . . . . . . 8 ⊢ (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))} ↔ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))}))
65	64	adantl 487	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))} ↔ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))}))
66	65	orbi2d 929	. . . . . 6 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → ((⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))}) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))})))
67		eqid 2762	. . . . . . . . . 10 ⊢ (∅ Sat ∅) = (∅ Sat ∅)
68	67	satf0suc 35963	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → ((∅ Sat ∅)‘suc 𝑦) = (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))}))
69	68	eleq2d 2848	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))})))
70		elun 4103	. . . . . . . 8 ⊢ (⟨𝐹, ∅⟩ ∈ (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))}) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))}))
71	69, 70	bitrdi 290	. . . . . . 7 ⊢ (𝑦 ∈ ω → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))})))
72	71	ad2antrr 739	. . . . . 6 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))})))
73		fmlasuc0 35971	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (Fmla‘suc 𝑦) = ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))}))
74	73	eleq2d 2848	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ 𝐹 ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))})))
75	74	ad2antrr 739	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ 𝐹 ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))})))
76		elun 4103	. . . . . . . 8 ⊢ (𝐹 ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))}) ↔ (𝐹 ∈ (Fmla‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))}))
77	76	a1i 11	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))}) ↔ (𝐹 ∈ (Fmla‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))})))
78		simpr 490	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) → (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦))))
79	78	imp 412	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))
80	79	orbi1d 930	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → ((𝐹 ∈ (Fmla‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))}) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))})))
81	75, 77, 80	3bitrd 308	. . . . . 6 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))})))
82	66, 72, 81	3bitr4rd 315	. . . . 5 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦)))
83	82	exp31 425	. . . 4 ⊢ (𝑦 ∈ ω → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦))) → (𝐹 ∈ V → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦)))))
84	6, 12, 18, 24, 48, 83	finds 7897	. . 3 ⊢ (𝑁 ∈ ω → (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
85	84	com12 33	. 2 ⊢ (𝐹 ∈ V → (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
86		prcnel 3478	. . . . 5 ⊢ (¬ 𝐹 ∈ V → ¬ 𝐹 ∈ (Fmla‘𝑁))
87	86	adantr 486	. . . 4 ⊢ ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → ¬ 𝐹 ∈ (Fmla‘𝑁))
88		opprc1 4860	. . . . . 6 ⊢ (¬ 𝐹 ∈ V → ⟨𝐹, ∅⟩ = ∅)
89	88	adantr 486	. . . . 5 ⊢ ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → ⟨𝐹, ∅⟩ = ∅)
90		satf0n0 35965	. . . . . . 7 ⊢ (𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))
91		df-nel 3064	. . . . . . 7 ⊢ (∅ ∉ ((∅ Sat ∅)‘𝑁) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
92	90, 91	sylib 221	. . . . . 6 ⊢ (𝑁 ∈ ω → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
93	92	adantl 487	. . . . 5 ⊢ ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
94	89, 93	eqneltrd 2882	. . . 4 ⊢ ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → ¬ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
95	87, 94	2falsed 379	. . 3 ⊢ ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
96	95	ex 418	. 2 ⊢ (¬ 𝐹 ∈ V → (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
97	85, 96	pm2.61i 184	1 ⊢ (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 {cab 2740 ∉ wnel 3063 ∃wrex 3088 {crab 3414 Vcvv 3453 ∪ cun 3900 ∅c0 4282 ⟨cop 4593 {copab 5171 suc csuc 6363 ‘cfv 6537 (class class class)co 7417 ωcom 7866 1^st c1st 7988 ∈_𝑔cgoe 35920 ⊼_𝑔cgna 35921 ∀_𝑔cgol 35922 Sat csat 35923 Fmlacfmla 35924

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-map 8832 df-goel 35927 df-sat 35930 df-fmla 35932

This theorem is used by: fmlasuc 35973

W3C validator