Theorem fmla0disjsuc 35366

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 35350	. . . 4 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)}
2		rabab 3520	. . . 4 ⊢ {𝑥 ∈ V ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)} = {𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)}
3	1, 2	eqtri 2768	. . 3 ⊢ (Fmla‘∅) = {𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)}
4	3	ineq1i 4237	. 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 4328	. . 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 35315	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑗∈𝑔𝑘) = ⟨∅, ⟨𝑗, 𝑘⟩⟩)
7	6	eqeq2d 2751	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗∈𝑔𝑘) ↔ 𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩))
8		1n0 8544	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1o ≠ ∅
9	8	nesymi 3004	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ ∅ = 1o
10	9	intnanr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩)
11		gonafv 35318	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 ∈ V ∧ 𝑣 ∈ V) → (𝑢⊼𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
12	11	el2v 3495	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢⊼𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩
13	12	eqeq2i 2753	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢⊼𝑔𝑣) ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
14		0ex 5325	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ∈ V
15		opex 5484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
16	14, 15	opth 5496	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨1o, ⟨𝑢, 𝑣⟩⟩ ↔ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩))
17	13, 16	bitri 275	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢⊼𝑔𝑣) ↔ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩))
18	10, 17	mtbir 323	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢⊼𝑔𝑣)
19		eqeq1 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = (𝑢⊼𝑔𝑣) ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢⊼𝑔𝑣)))
20	18, 19	mtbiri 327	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → ¬ 𝑥 = (𝑢⊼𝑔𝑣))
21	7, 20	biimtrdi 253	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗∈𝑔𝑘) → ¬ 𝑥 = (𝑢⊼𝑔𝑣)))
22	21	imp 406	. . . . . . . . . . . . . 14 ⊢ (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) → ¬ 𝑥 = (𝑢⊼𝑔𝑣))
23	22	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ¬ 𝑥 = (𝑢⊼𝑔𝑣))
24	23	ralrimivw 3156	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣))
25		2on0 8538	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2o ≠ ∅
26	25	nesymi 3004	. . . . . . . . . . . . . . . . . . . 20 ⊢ ¬ ∅ = 2o
27	26	orci 864	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩)
28	14, 15	opth 5496	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (∅ = 2o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
29	28	notbii 320	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ ¬ (∅ = 2o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
30		ianor 982	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ (∅ = 2o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩) ↔ (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
31	29, 30	bitri 275	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
32	27, 31	mpbir 231	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩
33		eqeq1 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ∀𝑔𝑖𝑢))
34		df-goal 35310	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∀𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
35	34	eqeq2i 2753	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
36	33, 35	bitrdi 287	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩))
37	32, 36	mtbiri 327	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → ¬ 𝑥 = ∀𝑔𝑖𝑢)
38	7, 37	biimtrdi 253	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗∈𝑔𝑘) → ¬ 𝑥 = ∀𝑔𝑖𝑢))
39	38	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) → ¬ 𝑥 = ∀𝑔𝑖𝑢)
40	39	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ¬ 𝑥 = ∀𝑔𝑖𝑢)
41	40	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) ∧ 𝑖 ∈ ω) → ¬ 𝑥 = ∀𝑔𝑖𝑢)
42	41	ralrimiva 3152	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢)
43	24, 42	jca 511	. . . . . . . . . . 11 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → (∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢))
44	43	ralrimiva 3152	. . . . . . . . . 10 ⊢ (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) → ∀𝑢 ∈ (Fmla‘∅)(∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢))
45		ralnex 3078	. . . . . . . . . . . . . 14 ⊢ (∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ↔ ¬ ∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣))
46		ralnex 3078	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢 ↔ ¬ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)
47	45, 46	anbi12i 627	. . . . . . . . . . . . 13 ⊢ ((∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
48		ioran 984	. . . . . . . . . . . . 13 ⊢ (¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
49	47, 48	bitr4i 278	. . . . . . . . . . . 12 ⊢ ((∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢) ↔ ¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
50	49	ralbii 3099	. . . . . . . . . . 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 3078	. . . . . . . . . . 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 275	. . . . . . . . . 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 218	. . . . . . . . 9 ⊢ (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
54	53	ex 412	. . . . . . . 8 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗∈𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
55	54	rexlimdva 3161	. . . . . . 7 ⊢ (𝑗 ∈ ω → (∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
56	55	rexlimiv 3154	. . . . . 6 ⊢ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
57	56	imori 853	. . . . 5 ⊢ (¬ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) ∨ ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
58		ianor 982	. . . . 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 231	. . . 4 ⊢ ¬ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
60	59	abf 4429	. . 3 ⊢ {𝑥 ∣ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))} = ∅
61	5, 60	eqtri 2768	. 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 2768	1 ⊢ ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488   ∩ cin 3975  ∅c0 4352  ⟨cop 4654  ‘cfv 6573  (class class class)co 7448  ωcom 7903  1_oc1o 8515  2_oc2o 8516  ∈_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-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-2o 8523  df-map 8886  df-goel 35308  df-gona 35309  df-goal 35310  df-sat 35311  df-fmla 35313

This theorem is referenced by:  satffunlem1lem2  35371