fmlasuc - Mathbox for Mario Carneiro

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Theorem fmlasuc 34675

Description: The valid Godel formulas of height (𝑁 + 1), expressed by the valid Godel formulas of height 𝑁. (Contributed by AV, 20-Sep-2023.)

**Assertion**
Ref	Expression
fmlasuc	⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)}))

Distinct variable group: 𝑢,𝑁,𝑣,𝑥,𝑖

Proof of Theorem fmlasuc Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmlasuc0 34673	. 2 ⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦))}))
2		eqid 2730	. . . . . . . 8 ⊢ (∅ Sat ∅) = (∅ Sat ∅)
3	2	satf0op 34666	. . . . . . 7 ⊢ (𝑁 ∈ ω → (𝑦 ∈ ((∅ Sat ∅)‘𝑁) ↔ ∃𝑧(𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
4		fveq2 6890	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (1^st ‘𝑧) = (1^st ‘𝑤))
5	4	oveq2d 7427	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)))
6	5	eqeq2d 2741	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ↔ 𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤))))
7	6	cbvrexvw 3233	. . . . . . . . 9 ⊢ (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ↔ ∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)))
8	7	orbi1i 910	. . . . . . . 8 ⊢ ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)) ↔ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)))
9		fmlafvel 34674	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (𝑧 ∈ (Fmla‘𝑁) ↔ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
10	9	biimprd 247	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → (⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁) → 𝑧 ∈ (Fmla‘𝑁)))
11	10	adantld 489	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → ((𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → 𝑧 ∈ (Fmla‘𝑁)))
12	11	imp 405	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → 𝑧 ∈ (Fmla‘𝑁))
13		vex 3476	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
14		0ex 5306	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
15	13, 14	op1std 7987	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (1^st ‘𝑦) = 𝑧)
16	15	eleq1d 2816	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((1^st ‘𝑦) ∈ (Fmla‘𝑁) ↔ 𝑧 ∈ (Fmla‘𝑁)))
17	16	ad2antrl 724	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((1^st ‘𝑦) ∈ (Fmla‘𝑁) ↔ 𝑧 ∈ (Fmla‘𝑁)))
18	12, 17	mpbird 256	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (1^st ‘𝑦) ∈ (Fmla‘𝑁))
19	18	3adant3 1130	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) ∧ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦))) → (1^st ‘𝑦) ∈ (Fmla‘𝑁))
20		oveq1 7418	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (1^st ‘𝑦) → (𝑢⊼_𝑔𝑣) = ((1^st ‘𝑦)⊼_𝑔𝑣))
21	20	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (1^st ‘𝑦) → (𝑥 = (𝑢⊼_𝑔𝑣) ↔ 𝑥 = ((1^st ‘𝑦)⊼_𝑔𝑣)))
22	21	rexbidv 3176	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1^st ‘𝑦) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔𝑣)))
23		eqidd 2731	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (1^st ‘𝑦) → 𝑖 = 𝑖)
24		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (1^st ‘𝑦) → 𝑢 = (1^st ‘𝑦))
25	23, 24	goaleq12d 34640	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (1^st ‘𝑦) → ∀_𝑔𝑖𝑢 = ∀_𝑔𝑖(1^st ‘𝑦))
26	25	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (1^st ‘𝑦) → (𝑥 = ∀_𝑔𝑖𝑢 ↔ 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)))
27	26	rexbidv 3176	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1^st ‘𝑦) → (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)))
28	22, 27	orbi12d 915	. . . . . . . . . . . 12 ⊢ (𝑢 = (1^st ‘𝑦) → ((∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦))))
29	28	adantl 480	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) ∧ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦))) ∧ 𝑢 = (1^st ‘𝑦)) → ((∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦))))
30	2	satf0op 34666	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ω → (𝑤 ∈ ((∅ Sat ∅)‘𝑁) ↔ ∃𝑦(𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
31		fmlafvel 34674	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ω → (𝑦 ∈ (Fmla‘𝑁) ↔ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
32	31	biimprd 247	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ω → (⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁) → 𝑦 ∈ (Fmla‘𝑁)))
33	32	adantld 489	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ω → ((𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → 𝑦 ∈ (Fmla‘𝑁)))
34	33	imp 405	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → 𝑦 ∈ (Fmla‘𝑁))
35		vex 3476	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑦 ∈ V
36	35, 14	op1std 7987	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = ⟨𝑦, ∅⟩ → (1^st ‘𝑤) = 𝑦)
37	36	eleq1d 2816	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = ⟨𝑦, ∅⟩ → ((1^st ‘𝑤) ∈ (Fmla‘𝑁) ↔ 𝑦 ∈ (Fmla‘𝑁)))
38	37	ad2antrl 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((1^st ‘𝑤) ∈ (Fmla‘𝑁) ↔ 𝑦 ∈ (Fmla‘𝑁)))
39	34, 38	mpbird 256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (1^st ‘𝑤) ∈ (Fmla‘𝑁))
40	39	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤))) → (1^st ‘𝑤) ∈ (Fmla‘𝑁))
41		oveq2 7419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (1^st ‘𝑤) → (𝑧⊼_𝑔𝑣) = (𝑧⊼_𝑔(1^st ‘𝑤)))
42	41	eqeq2d 2741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (1^st ‘𝑤) → (𝑥 = (𝑧⊼_𝑔𝑣) ↔ 𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤))))
43	42	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤))) ∧ 𝑣 = (1^st ‘𝑤)) → (𝑥 = (𝑧⊼_𝑔𝑣) ↔ 𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤))))
44		simpr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤))) → 𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤)))
45	40, 43, 44	rspcedvd 3613	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤))) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼_𝑔𝑣))
46	45	exp31 418	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ω → ((𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → (𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼_𝑔𝑣))))
47	46	exlimdv 1934	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ω → (∃𝑦(𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → (𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼_𝑔𝑣))))
48	30, 47	sylbid 239	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (𝑤 ∈ ((∅ Sat ∅)‘𝑁) → (𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼_𝑔𝑣))))
49	48	rexlimdv 3151	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼_𝑔𝑣)))
50	49	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼_𝑔𝑣)))
51	15	oveq1d 7426	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) = (𝑧⊼_𝑔(1^st ‘𝑤)))
52	51	eqeq2d 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) ↔ 𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤))))
53	52	rexbidv 3176	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) ↔ ∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤))))
54	15	oveq1d 7426	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((1^st ‘𝑦)⊼_𝑔𝑣) = (𝑧⊼_𝑔𝑣))
55	54	eqeq2d 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (𝑥 = ((1^st ‘𝑦)⊼_𝑔𝑣) ↔ 𝑥 = (𝑧⊼_𝑔𝑣)))
56	55	rexbidv 3176	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼_𝑔𝑣)))
57	53, 56	imbi12d 343	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔𝑣)) ↔ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼_𝑔𝑣))))
58	57	ad2antrl 724	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔𝑣)) ↔ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼_𝑔(1^st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼_𝑔𝑣))))
59	50, 58	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔𝑣)))
60	59	orim1d 962	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦))))
61	60	3impia 1115	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) ∧ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦))) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)))
62	19, 29, 61	rspcedvd 3613	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) ∧ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦))) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢))
63	62	3exp 1117	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → ((𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢))))
64	63	exlimdv 1934	. . . . . . . 8 ⊢ (𝑁 ∈ ω → (∃𝑧(𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢))))
65	8, 64	syl7bi 254	. . . . . . 7 ⊢ (𝑁 ∈ ω → (∃𝑧(𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢))))
66	3, 65	sylbid 239	. . . . . 6 ⊢ (𝑁 ∈ ω → (𝑦 ∈ ((∅ Sat ∅)‘𝑁) → ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢))))
67	66	rexlimdv 3151	. . . . 5 ⊢ (𝑁 ∈ ω → (∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)))
68		fmlafvel 34674	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → (𝑢 ∈ (Fmla‘𝑁) ↔ ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
69	68	biimpa 475	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
70	69	adantr 479	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)) → ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
71		vex 3476	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
72	71, 14	op1std 7987	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (1^st ‘𝑦) = 𝑢)
73	72	oveq1d 7426	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) = (𝑢⊼_𝑔(1^st ‘𝑧)))
74	73	eqeq2d 2741	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ↔ 𝑥 = (𝑢⊼_𝑔(1^st ‘𝑧))))
75	74	rexbidv 3176	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ↔ ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼_𝑔(1^st ‘𝑧))))
76		eqidd 2731	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → 𝑖 = 𝑖)
77	76, 72	goaleq12d 34640	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → ∀_𝑔𝑖(1^st ‘𝑦) = ∀_𝑔𝑖𝑢)
78	77	eqeq2d 2741	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (𝑥 = ∀_𝑔𝑖(1^st ‘𝑦) ↔ 𝑥 = ∀_𝑔𝑖𝑢))
79	78	rexbidv 3176	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦) ↔ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢))
80	75, 79	orbi12d 915	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)) ↔ (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)))
81	80	adantl 480	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)) ∧ 𝑦 = ⟨𝑢, ∅⟩) → ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)) ↔ (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)))
82		fmlafvel 34674	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → (𝑣 ∈ (Fmla‘𝑁) ↔ ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
83	82	biimpd 228	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → (𝑣 ∈ (Fmla‘𝑁) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
84	83	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → (𝑣 ∈ (Fmla‘𝑁) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
85	84	imp 405	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
86	85	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼_𝑔𝑣)) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
87		vex 3476	. . . . . . . . . . . . . . 15 ⊢ 𝑣 ∈ V
88	87, 14	op1std 7987	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑣, ∅⟩ → (1^st ‘𝑧) = 𝑣)
89	88	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑣, ∅⟩ → (𝑢⊼_𝑔(1^st ‘𝑧)) = (𝑢⊼_𝑔𝑣))
90	89	eqeq2d 2741	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑣, ∅⟩ → (𝑥 = (𝑢⊼_𝑔(1^st ‘𝑧)) ↔ 𝑥 = (𝑢⊼_𝑔𝑣)))
91	90	adantl 480	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼_𝑔𝑣)) ∧ 𝑧 = ⟨𝑣, ∅⟩) → (𝑥 = (𝑢⊼_𝑔(1^st ‘𝑧)) ↔ 𝑥 = (𝑢⊼_𝑔𝑣)))
92		simpr 483	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼_𝑔𝑣)) → 𝑥 = (𝑢⊼_𝑔𝑣))
93	86, 91, 92	rspcedvd 3613	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼_𝑔𝑣)) → ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼_𝑔(1^st ‘𝑧)))
94	93	rexlimdva2 3155	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) → ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼_𝑔(1^st ‘𝑧))))
95	94	orim1d 962	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → ((∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢) → (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)))
96	95	imp 405	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)) → (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢))
97	70, 81, 96	rspcedvd 3613	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)) → ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)))
98	97	rexlimdva2 3155	. . . . 5 ⊢ (𝑁 ∈ ω → (∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢) → ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦))))
99	67, 98	impbid 211	. . . 4 ⊢ (𝑁 ∈ ω → (∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦)) ↔ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)))
100	99	abbidv 2799	. . 3 ⊢ (𝑁 ∈ ω → {𝑥 ∣ ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦))} = {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)})
101	100	uneq2d 4162	. 2 ⊢ (𝑁 ∈ ω → ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1^st ‘𝑦)⊼_𝑔(1^st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑦))}) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)}))
102	1, 101	eqtrd 2770	1 ⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖𝑢)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∃wex 1779 ∈ wcel 2104 {cab 2707 ∃wrex 3068 ∪ cun 3945 ∅c0 4321 ⟨cop 4633 suc csuc 6365 ‘cfv 6542 (class class class)co 7411 ωcom 7857 1^st c1st 7975 ⊼_𝑔cgna 34623 ∀_𝑔cgol 34624 Sat csat 34625 Fmlacfmla 34626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-map 8824 df-goel 34629 df-goal 34631 df-sat 34632 df-fmla 34634

This theorem is referenced by: fmla1 34676 isfmlasuc 34677 fmlasssuc 34678 fmlaomn0 34679

W3C validator