Theorem 2goelgoanfmla1 36086

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 7424	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐿 → (𝐾∈𝑔𝑛) = (𝐾∈𝑔𝐿))
6	5	oveq2d 7432	. . . . . . . . . 10 ⊢ (𝑛 = 𝐿 → ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿)))
7	6	eqeq2d 2771	. . . . . . . . 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 3578	. . . . . . 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 7423	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑖∈𝑔𝑗) = (𝐼∈𝑔𝑗))
14	13	oveq1d 7431	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)))
15	14	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
16	15	rexbidv 3186	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
17		eqidd 2761	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → 𝑘 = 𝑘)
18	17, 13	goaleq12d 36013	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ∀𝑔𝑘(𝑖∈𝑔𝑗) = ∀𝑔𝑘(𝐼∈𝑔𝑗))
19	18	eqeq2d 2771	. . . . . . . 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 7424	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝐼∈𝑔𝑗) = (𝐼∈𝑔𝐽))
22	21	oveq1d 7431	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)))
23	22	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛))))
24	23	rexbidv 3186	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛))))
25		eqidd 2761	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → 𝑘 = 𝑘)
26	25, 21	goaleq12d 36013	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → ∀𝑔𝑘(𝐼∈𝑔𝑗) = ∀𝑔𝑘(𝐼∈𝑔𝐽))
27	26	eqeq2d 2771	. . . . . . . 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 7423	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘∈𝑔𝑛) = (𝐾∈𝑔𝑛))
30	29	oveq2d 7432	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)))
31	30	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛))))
32	31	rexbidv 3186	. . . . . . . 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 2761	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝐼∈𝑔𝐽) = (𝐼∈𝑔𝐽))
35	33, 34	goaleq12d 36013	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ∀𝑔𝑘(𝐼∈𝑔𝐽) = ∀𝑔𝐾(𝐼∈𝑔𝐽))
36	35	eqeq2d 2771	. . . . . . . 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 3593	. . . . . 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 1400	. . . . 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 7450	. . . . . 6 ⊢ 𝑋 ∈ V
41		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
42	41	rexbidv 3186	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
43		eqeq1 2764	. . . . . . . . 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 3186	. . . . . . 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 3227	. . . . . 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 3633	. . . . 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 4100	. . 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 36049	. 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 2871	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 2145  {cab 2738  ∃wrex 3086   ∪ cun 3897  ∅c0 4279  {csn 4584   × cxp 5653  ‘cfv 6535  (class class class)co 7416  ωcom 7868  1_oc1o 8455  ∈_𝑔cgoe 35995  ⊼_𝑔cgna 35996  ∀_𝑔cgol 35997  Fmlacfmla 35999

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 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-inf2 9627

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-map 8835  df-goel 36002  df-goal 36004  df-sat 36005  df-fmla 36007

This theorem is used by:  satefvfmla1  36087