Theorem 2goelgoanfmla1 36355

Description: Two Godel-sets of membership combined with a Godel-set for NAND is a Godel formula of height 1. (Contributed by AV, 17-Nov-2023.)

Hypothesis

Ref	Expression
satfv1fvfmla1.x	⊢ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿))

Assertion

Ref	Expression
2goelgoanfmla1	⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))

Proof of Theorem 2goelgoanfmla1

Dummy variables 𝑖 𝑗 𝑘 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 779	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐼 ∈ ω)
2		simplr 781	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐽 ∈ ω)
3		simprl 783	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐾 ∈ ω)
4		simprr 785	. . . . . . . 8 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐿 ∈ ω)
5		oveq2 7448	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐿 → (𝐾∈𝑔𝑛) = (𝐾∈𝑔𝐿))
6	5	oveq2d 7456	. . . . . . . . . 10 ⊢ (𝑛 = 𝐿 → ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿)))
7	6	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑛 = 𝐿 → (𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿))))
8	7	adantl 487	. . . . . . . 8 ⊢ ((((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑛 = 𝐿) → (𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿))))
9		satfv1fvfmla1.x	. . . . . . . . 9 ⊢ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿))
10	9	a1i 11	. . . . . . . 8 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿)))
11	4, 8, 10	rspcedvd 3582	. . . . . . 7 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)))
12	11	orcd 887	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝐾(𝐼∈𝑔𝐽)))
13		oveq1 7447	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑖∈𝑔𝑗) = (𝐼∈𝑔𝑗))
14	13	oveq1d 7455	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)))
15	14	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
16	15	rexbidv 3190	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
17		eqidd 2764	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → 𝑘 = 𝑘)
18	17, 13	goaleq12d 36282	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ∀𝑔𝑘(𝑖∈𝑔𝑗) = ∀𝑔𝑘(𝐼∈𝑔𝑗))
19	18	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝑗)))
20	16, 19	orbi12d 932	. . . . . . 7 ⊢ (𝑖 = 𝐼 → ((∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝑗))))
21		oveq2 7448	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝐼∈𝑔𝑗) = (𝐼∈𝑔𝐽))
22	21	oveq1d 7455	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)))
23	22	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛))))
24	23	rexbidv 3190	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛))))
25		eqidd 2764	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → 𝑘 = 𝑘)
26	25, 21	goaleq12d 36282	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → ∀𝑔𝑘(𝐼∈𝑔𝑗) = ∀𝑔𝑘(𝐼∈𝑔𝐽))
27	26	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝐽)))
28	24, 27	orbi12d 932	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ((∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝑗)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝐽))))
29		oveq1 7447	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘∈𝑔𝑛) = (𝐾∈𝑔𝑛))
30	29	oveq2d 7456	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)))
31	30	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛))))
32	31	rexbidv 3190	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛))))
33		id 23	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → 𝑘 = 𝐾)
34		eqidd 2764	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝐼∈𝑔𝐽) = (𝐼∈𝑔𝐽))
35	33, 34	goaleq12d 36282	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ∀𝑔𝑘(𝐼∈𝑔𝐽) = ∀𝑔𝐾(𝐼∈𝑔𝐽))
36	35	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝐽) ↔ 𝑋 = ∀𝑔𝐾(𝐼∈𝑔𝐽)))
37	32, 36	orbi12d 932	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝐽)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝐾(𝐼∈𝑔𝐽))))
38	20, 28, 37	rspc3ev 3596	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐾 ∈ ω) ∧ (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝐾(𝐼∈𝑔𝐽))) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
39	1, 2, 3, 12, 38	syl31anc 1401	. . . . 5 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
40	9	ovexi 7474	. . . . . 6 ⊢ 𝑋 ∈ V
41		eqeq1 2767	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
42	41	rexbidv 3190	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
43		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
44	42, 43	orbi12d 932	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗))))
45	44	rexbidv 3190	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)) ↔ ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗))))
46	45	2rexbidv 3231	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗))))
47	40, 46	elab 3636	. . . . 5 ⊢ (𝑋 ∈ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))} ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
48	39, 47	sylibr 237	. . . 4 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))})
49	48	olcd 888	. . 3 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑋 ∈ ({∅} × (ω × ω)) ∨ 𝑋 ∈ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))}))
50		elun 4155	. . 3 ⊢ (𝑋 ∈ (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))}) ↔ (𝑋 ∈ ({∅} × (ω × ω)) ∨ 𝑋 ∈ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))}))
51	49, 50	sylibr 237	. 2 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))}))
52		fmla1 36318	. 2 ⊢ (Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))})
53	51, 52	eleqtrrdi 2875	1 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090   ∪ cun 4152  ∅c0 4281  {csn 4588   × cxp 5653  ‘cfv 6555  (class class class)co 7440  ωcom 7898  1_oc1o 8494  ∈_𝑔cgoe 36264  ⊼_𝑔cgna 36265  ∀_𝑔cgol 36266  Fmlacfmla 36268

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-inf2 9666

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-map 8874  df-goel 36271  df-goal 36273  df-sat 36274  df-fmla 36276

This theorem is used by:  satefvfmla1  36356