Theorem 2goelgoanfmla1 35392

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 766	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐼 ∈ ω)
2		simplr 768	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐽 ∈ ω)
3		simprl 770	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐾 ∈ ω)
4		simprr 772	. . . . . . . 8 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐿 ∈ ω)
5		oveq2 7456	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐿 → (𝐾∈𝑔𝑛) = (𝐾∈𝑔𝐿))
6	5	oveq2d 7464	. . . . . . . . . 10 ⊢ (𝑛 = 𝐿 → ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿)))
7	6	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑛 = 𝐿 → (𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿))))
8	7	adantl 481	. . . . . . . 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 3637	. . . . . . 7 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)))
12	11	orcd 872	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝐾(𝐼∈𝑔𝐽)))
13		oveq1 7455	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑖∈𝑔𝑗) = (𝐼∈𝑔𝑗))
14	13	oveq1d 7463	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)))
15	14	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
16	15	rexbidv 3185	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
17		eqidd 2741	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → 𝑘 = 𝑘)
18	17, 13	goaleq12d 35319	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ∀𝑔𝑘(𝑖∈𝑔𝑗) = ∀𝑔𝑘(𝐼∈𝑔𝑗))
19	18	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝑗)))
20	16, 19	orbi12d 917	. . . . . . 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 7456	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝐼∈𝑔𝑗) = (𝐼∈𝑔𝐽))
22	21	oveq1d 7463	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)))
23	22	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛))))
24	23	rexbidv 3185	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛))))
25		eqidd 2741	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → 𝑘 = 𝑘)
26	25, 21	goaleq12d 35319	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → ∀𝑔𝑘(𝐼∈𝑔𝑗) = ∀𝑔𝑘(𝐼∈𝑔𝐽))
27	26	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝐽)))
28	24, 27	orbi12d 917	. . . . . . 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 7455	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘∈𝑔𝑛) = (𝐾∈𝑔𝑛))
30	29	oveq2d 7464	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)))
31	30	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛))))
32	31	rexbidv 3185	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛))))
33		id 22	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → 𝑘 = 𝐾)
34		eqidd 2741	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝐼∈𝑔𝐽) = (𝐼∈𝑔𝐽))
35	33, 34	goaleq12d 35319	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ∀𝑔𝑘(𝐼∈𝑔𝐽) = ∀𝑔𝐾(𝐼∈𝑔𝐽))
36	35	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝐽) ↔ 𝑋 = ∀𝑔𝐾(𝐼∈𝑔𝐽)))
37	32, 36	orbi12d 917	. . . . . . 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 3652	. . . . . 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 1373	. . . . 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 7482	. . . . . 6 ⊢ 𝑋 ∈ V
41		eqeq1 2744	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
42	41	rexbidv 3185	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
43		eqeq1 2744	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
44	42, 43	orbi12d 917	. . . . . . . 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 3185	. . . . . . 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 3228	. . . . . 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 3694	. . . . 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 234	. . . 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 873	. . 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 4176	. . 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 234	. 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 35355	. 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 2855	1 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076   ∪ cun 3974  ∅c0 4352  {csn 4648   × cxp 5698  ‘cfv 6573  (class class class)co 7448  ωcom 7903  1_oc1o 8515  ∈_𝑔cgoe 35301  ⊼_𝑔cgna 35302  ∀_𝑔cgol 35303  Fmlacfmla 35305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-map 8886  df-goel 35308  df-goal 35310  df-sat 35311  df-fmla 35313

This theorem is referenced by:  satefvfmla1  35393