Theorem fmla0disjsuc 36132

Description: The set of valid Godel formulas of height 0 is disjoint with the formulas constructed from Godel-sets for the Sheffer stroke NAND and Godel-set of universal quantification. (Contributed by AV, 20-Oct-2023.)

Assertion

Ref	Expression
fmla0disjsuc	⊢ ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅

Distinct variable group:   𝑢,𝑖,𝑣,𝑥

Proof of Theorem fmla0disjsuc

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

Step	Hyp	Ref	Expression
1		fmla0 36116	. . . 4 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)}
2		rabab 3481	. . . 4 ⊢ {𝑥 ∈ V ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)} = {𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)}
3	1, 2	eqtri 2784	. . 3 ⊢ (Fmla‘∅) = {𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)}
4	3	ineq1i 4162	. 2 ⊢ ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ({𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)} ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)})
5		inab 4255	. . 3 ⊢ ({𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)} ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = {𝑥 ∣ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))}
6		goel 36081	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑗∈𝑔𝑘) = ⟨∅, ⟨𝑗, 𝑘⟩⟩)
7	6	eqeq2d 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗∈𝑔𝑘) ↔ 𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩))
8		1n0 8479	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1o ≠ ∅
9	8	nesymi 3013	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ ∅ = 1o
10	9	intnanr 493	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩)
11		gonafv 36084	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 ∈ V ∧ 𝑣 ∈ V) → (𝑢⊼𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
12	11	el2v 3458	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢⊼𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩
13	12	eqeq2i 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢⊼𝑔𝑣) ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
14		0ex 5261	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ∈ V
15		opex 5432	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
16	14, 15	opth 5445	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨1o, ⟨𝑢, 𝑣⟩⟩ ↔ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩))
17	13, 16	bitri 278	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢⊼𝑔𝑣) ↔ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩))
18	10, 17	mtbir 326	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢⊼𝑔𝑣)
19		eqeq1 2765	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = (𝑢⊼𝑔𝑣) ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢⊼𝑔𝑣)))
20	18, 19	mtbiri 330	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → ¬ 𝑥 = (𝑢⊼𝑔𝑣))
21	7, 20	biimtrdi 256	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗∈𝑔𝑘) → ¬ 𝑥 = (𝑢⊼𝑔𝑣)))
22	21	imp 412	. . . . . . . . . . . . . 14 ⊢ (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) → ¬ 𝑥 = (𝑢⊼𝑔𝑣))
23	22	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ¬ 𝑥 = (𝑢⊼𝑔𝑣))
24	23	ralrimivw 3159	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣))
25		2on0 8475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2o ≠ ∅
26	25	nesymi 3013	. . . . . . . . . . . . . . . . . . . 20 ⊢ ¬ ∅ = 2o
27	26	orci 879	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩)
28	14, 15	opth 5445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (∅ = 2o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
29	28	notbii 323	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ ¬ (∅ = 2o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
30		ianor 997	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ (∅ = 2o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩) ↔ (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
31	29, 30	bitri 278	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
32	27, 31	mpbir 234	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩
33		eqeq1 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ∀𝑔𝑖𝑢))
34		df-goal 36076	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∀𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
35	34	eqeq2i 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
36	33, 35	bitrdi 290	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩))
37	32, 36	mtbiri 330	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → ¬ 𝑥 = ∀𝑔𝑖𝑢)
38	7, 37	biimtrdi 256	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗∈𝑔𝑘) → ¬ 𝑥 = ∀𝑔𝑖𝑢))
39	38	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) → ¬ 𝑥 = ∀𝑔𝑖𝑢)
40	39	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ¬ 𝑥 = ∀𝑔𝑖𝑢)
41	40	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) ∧ 𝑖 ∈ ω) → ¬ 𝑥 = ∀𝑔𝑖𝑢)
42	41	ralrimiva 3155	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢)
43	24, 42	jca 521	. . . . . . . . . . 11 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → (∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢))
44	43	ralrimiva 3155	. . . . . . . . . 10 ⊢ (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) → ∀𝑢 ∈ (Fmla‘∅)(∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢))
45		ralnex 3089	. . . . . . . . . . . . . 14 ⊢ (∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ↔ ¬ ∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣))
46		ralnex 3089	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢 ↔ ¬ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)
47	45, 46	anbi12i 640	. . . . . . . . . . . . 13 ⊢ ((∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
48		ioran 999	. . . . . . . . . . . . 13 ⊢ (¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
49	47, 48	bitr4i 281	. . . . . . . . . . . 12 ⊢ ((∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢) ↔ ¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
50	49	ralbii 3109	. . . . . . . . . . 11 ⊢ (∀𝑢 ∈ (Fmla‘∅)(∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢) ↔ ∀𝑢 ∈ (Fmla‘∅) ¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
51		ralnex 3089	. . . . . . . . . . 11 ⊢ (∀𝑢 ∈ (Fmla‘∅) ¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
52	50, 51	bitri 278	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ (Fmla‘∅)(∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢) ↔ ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
53	44, 52	sylib 221	. . . . . . . . 9 ⊢ (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
54	53	ex 418	. . . . . . . 8 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗∈𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
55	54	rexlimdva 3164	. . . . . . 7 ⊢ (𝑗 ∈ ω → (∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
56	55	rexlimiv 3157	. . . . . 6 ⊢ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
57	56	imori 868	. . . . 5 ⊢ (¬ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) ∨ ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
58		ianor 997	. . . . 5 ⊢ (¬ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)) ↔ (¬ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) ∨ ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
59	57, 58	mpbir 234	. . . 4 ⊢ ¬ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
60	59	abf 4364	. . 3 ⊢ {𝑥 ∣ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))} = ∅
61	5, 60	eqtri 2784	. 2 ⊢ ({𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)} ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅
62	4, 61	eqtri 2784	1 ⊢ ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451   ∩ cin 3898  ∅c0 4279  ⟨cop 4590  ‘cfv 6531  (class class class)co 7412  ωcom 7866  1_oc1o 8453  2_oc2o 8454  ∈_𝑔cgoe 36067  ⊼_𝑔cgna 36068  ∀_𝑔cgol 36069  Fmlacfmla 36071

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-map 8833  df-goel 36074  df-gona 36075  df-goal 36076  df-sat 36077  df-fmla 36079

This theorem is used by:  satffunlem1lem2  36137