Theorem 2goelgoanfmla1 33286

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 763	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐼 ∈ ω)
2		simplr 765	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐽 ∈ ω)
3		simprl 767	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐾 ∈ ω)
4		simprr 769	. . . . . . . 8 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐿 ∈ ω)
5		oveq2 7263	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐿 → (𝐾∈𝑔𝑛) = (𝐾∈𝑔𝐿))
6	5	oveq2d 7271	. . . . . . . . . 10 ⊢ (𝑛 = 𝐿 → ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿)))
7	6	eqeq2d 2749	. . . . . . . . 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 3555	. . . . . . 7 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)))
12	11	orcd 869	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝐾(𝐼∈𝑔𝐽)))
13		oveq1 7262	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑖∈𝑔𝑗) = (𝐼∈𝑔𝑗))
14	13	oveq1d 7270	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)))
15	14	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
16	15	rexbidv 3225	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
17		eqidd 2739	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → 𝑘 = 𝑘)
18	17, 13	goaleq12d 33213	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ∀𝑔𝑘(𝑖∈𝑔𝑗) = ∀𝑔𝑘(𝐼∈𝑔𝑗))
19	18	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝑗)))
20	16, 19	orbi12d 915	. . . . . . 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 7263	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝐼∈𝑔𝑗) = (𝐼∈𝑔𝐽))
22	21	oveq1d 7270	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)))
23	22	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛))))
24	23	rexbidv 3225	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛))))
25		eqidd 2739	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → 𝑘 = 𝑘)
26	25, 21	goaleq12d 33213	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → ∀𝑔𝑘(𝐼∈𝑔𝑗) = ∀𝑔𝑘(𝐼∈𝑔𝐽))
27	26	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝐽)))
28	24, 27	orbi12d 915	. . . . . . 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 7262	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘∈𝑔𝑛) = (𝐾∈𝑔𝑛))
30	29	oveq2d 7271	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)))
31	30	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛))))
32	31	rexbidv 3225	. . . . . . . 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 2739	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝐼∈𝑔𝐽) = (𝐼∈𝑔𝐽))
35	33, 34	goaleq12d 33213	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ∀𝑔𝑘(𝐼∈𝑔𝐽) = ∀𝑔𝐾(𝐼∈𝑔𝐽))
36	35	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝐽) ↔ 𝑋 = ∀𝑔𝐾(𝐼∈𝑔𝐽)))
37	32, 36	orbi12d 915	. . . . . . 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 3566	. . . . . 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 1371	. . . . 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 7289	. . . . . 6 ⊢ 𝑋 ∈ V
41		eqeq1 2742	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
42	41	rexbidv 3225	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
43		eqeq1 2742	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
44	42, 43	orbi12d 915	. . . . . . . 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 3225	. . . . . . 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 3602	. . . . 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 233	. . . 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 870	. . 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 4079	. . 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 233	. 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 33249	. 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 2850	1 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064   ∪ cun 3881  ∅c0 4253  {csn 4558   × cxp 5578  ‘cfv 6418  (class class class)co 7255  ωcom 7687  1_oc1o 8260  ∈_𝑔cgoe 33195  ⊼_𝑔cgna 33196  ∀_𝑔cgol 33197  Fmlacfmla 33199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-map 8575  df-goel 33202  df-goal 33204  df-sat 33205  df-fmla 33207

This theorem is referenced by:  satefvfmla1  33287