Theorem 2goelgoanfmla1 35626

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 767	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐼 ∈ ω)
2		simplr 769	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐽 ∈ ω)
3		simprl 771	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐾 ∈ ω)
4		simprr 773	. . . . . . . 8 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐿 ∈ ω)
5		oveq2 7370	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐿 → (𝐾∈𝑔𝑛) = (𝐾∈𝑔𝐿))
6	5	oveq2d 7378	. . . . . . . . . 10 ⊢ (𝑛 = 𝐿 → ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿)))
7	6	eqeq2d 2748	. . . . . . . . 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 3567	. . . . . . 7 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)))
12	11	orcd 874	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝐾(𝐼∈𝑔𝐽)))
13		oveq1 7369	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑖∈𝑔𝑗) = (𝐼∈𝑔𝑗))
14	13	oveq1d 7377	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)))
15	14	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
16	15	rexbidv 3162	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
17		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → 𝑘 = 𝑘)
18	17, 13	goaleq12d 35553	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ∀𝑔𝑘(𝑖∈𝑔𝑗) = ∀𝑔𝑘(𝐼∈𝑔𝑗))
19	18	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝑗)))
20	16, 19	orbi12d 919	. . . . . . 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 7370	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝐼∈𝑔𝑗) = (𝐼∈𝑔𝐽))
22	21	oveq1d 7377	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)))
23	22	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛))))
24	23	rexbidv 3162	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛))))
25		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → 𝑘 = 𝑘)
26	25, 21	goaleq12d 35553	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → ∀𝑔𝑘(𝐼∈𝑔𝑗) = ∀𝑔𝑘(𝐼∈𝑔𝐽))
27	26	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝐽)))
28	24, 27	orbi12d 919	. . . . . . 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 7369	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘∈𝑔𝑛) = (𝐾∈𝑔𝑛))
30	29	oveq2d 7378	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛)))
31	30	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝑛))))
32	31	rexbidv 3162	. . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝐼∈𝑔𝐽) = (𝐼∈𝑔𝐽))
35	33, 34	goaleq12d 35553	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ∀𝑔𝑘(𝐼∈𝑔𝐽) = ∀𝑔𝐾(𝐼∈𝑔𝐽))
36	35	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑋 = ∀𝑔𝑘(𝐼∈𝑔𝐽) ↔ 𝑋 = ∀𝑔𝐾(𝐼∈𝑔𝐽)))
37	32, 36	orbi12d 919	. . . . . . 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 3582	. . . . . 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 1376	. . . . 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 7396	. . . . . 6 ⊢ 𝑋 ∈ V
41		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
42	41	rexbidv 3162	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∃𝑛 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑛))))
43		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
44	42, 43	orbi12d 919	. . . . . . . 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 3162	. . . . . . 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 3203	. . . . . 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 3623	. . . . 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 875	. . 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 4094	. . 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 35589	. 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 2848	1 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062   ∪ cun 3888  ∅c0 4274  {csn 4568   × cxp 5624  ‘cfv 6494  (class class class)co 7362  ωcom 7812  1_oc1o 8393  ∈_𝑔cgoe 35535  ⊼_𝑔cgna 35536  ∀_𝑔cgol 35537  Fmlacfmla 35539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-inf2 9557

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-map 8770  df-goel 35542  df-goal 35544  df-sat 35545  df-fmla 35547

This theorem is referenced by:  satefvfmla1  35627