Theorem fmla0disjsuc 34079

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 34063	. . . 4 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)}
2		rabab 3474	. . . 4 ⊢ {𝑥 ∈ V ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)} = {𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)}
3	1, 2	eqtri 2759	. . 3 ⊢ (Fmla‘∅) = {𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘)}
4	3	ineq1i 4173	. 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 4264	. . 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 34028	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑗∈𝑔𝑘) = ⟨∅, ⟨𝑗, 𝑘⟩⟩)
7	6	eqeq2d 2742	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗∈𝑔𝑘) ↔ 𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩))
8		1n0 8439	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1o ≠ ∅
9	8	nesymi 2997	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ ∅ = 1o
10	9	intnanr 488	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩)
11		gonafv 34031	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 ∈ V ∧ 𝑣 ∈ V) → (𝑢⊼𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
12	11	el2v 3454	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢⊼𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩
13	12	eqeq2i 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢⊼𝑔𝑣) ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
14		0ex 5269	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ∈ V
15		opex 5426	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
16	14, 15	opth 5438	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨1o, ⟨𝑢, 𝑣⟩⟩ ↔ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩))
17	13, 16	bitri 274	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢⊼𝑔𝑣) ↔ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩))
18	10, 17	mtbir 322	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢⊼𝑔𝑣)
19		eqeq1 2735	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = (𝑢⊼𝑔𝑣) ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢⊼𝑔𝑣)))
20	18, 19	mtbiri 326	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → ¬ 𝑥 = (𝑢⊼𝑔𝑣))
21	7, 20	syl6bi 252	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗∈𝑔𝑘) → ¬ 𝑥 = (𝑢⊼𝑔𝑣)))
22	21	imp 407	. . . . . . . . . . . . . 14 ⊢ (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) → ¬ 𝑥 = (𝑢⊼𝑔𝑣))
23	22	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ¬ 𝑥 = (𝑢⊼𝑔𝑣))
24	23	ralrimivw 3143	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣))
25		2on0 8433	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2o ≠ ∅
26	25	nesymi 2997	. . . . . . . . . . . . . . . . . . . 20 ⊢ ¬ ∅ = 2o
27	26	orci 863	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩)
28	14, 15	opth 5438	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (∅ = 2o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
29	28	notbii 319	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ ¬ (∅ = 2o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
30		ianor 980	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ (∅ = 2o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩) ↔ (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
31	29, 30	bitri 274	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
32	27, 31	mpbir 230	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩
33		eqeq1 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ∀𝑔𝑖𝑢))
34		df-goal 34023	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∀𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
35	34	eqeq2i 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨∅, ⟨𝑗, 𝑘⟩⟩ = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
36	33, 35	bitrdi 286	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩))
37	32, 36	mtbiri 326	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → ¬ 𝑥 = ∀𝑔𝑖𝑢)
38	7, 37	syl6bi 252	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗∈𝑔𝑘) → ¬ 𝑥 = ∀𝑔𝑖𝑢))
39	38	imp 407	. . . . . . . . . . . . . . 15 ⊢ (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) → ¬ 𝑥 = ∀𝑔𝑖𝑢)
40	39	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ¬ 𝑥 = ∀𝑔𝑖𝑢)
41	40	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) ∧ 𝑖 ∈ ω) → ¬ 𝑥 = ∀𝑔𝑖𝑢)
42	41	ralrimiva 3139	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢)
43	24, 42	jca 512	. . . . . . . . . . 11 ⊢ ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → (∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢))
44	43	ralrimiva 3139	. . . . . . . . . 10 ⊢ (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) → ∀𝑢 ∈ (Fmla‘∅)(∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢))
45		ralnex 3071	. . . . . . . . . . . . . 14 ⊢ (∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ↔ ¬ ∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣))
46		ralnex 3071	. . . . . . . . . . . . . 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 982	. . . . . . . . . . . . 13 ⊢ (¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
49	47, 48	bitr4i 277	. . . . . . . . . . . 12 ⊢ ((∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢⊼𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢) ↔ ¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
50	49	ralbii 3092	. . . . . . . . . . 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 3071	. . . . . . . . . . 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 274	. . . . . . . . . 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 217	. . . . . . . . 9 ⊢ (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗∈𝑔𝑘)) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
54	53	ex 413	. . . . . . . 8 ⊢ ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗∈𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
55	54	rexlimdva 3148	. . . . . . 7 ⊢ (𝑗 ∈ ω → (∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
56	55	rexlimiv 3141	. . . . . 6 ⊢ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
57	56	imori 852	. . . . 5 ⊢ (¬ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) ∨ ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
58		ianor 980	. . . . 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 230	. . . 4 ⊢ ¬ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
60	59	abf 4367	. . 3 ⊢ {𝑥 ∣ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗∈𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))} = ∅
61	5, 60	eqtri 2759	. 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 2759	1 ⊢ ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  {cab 2708  ∀wral 3060  ∃wrex 3069  {crab 3405  Vcvv 3446   ∩ cin 3912  ∅c0 4287  ⟨cop 4597  ‘cfv 6501  (class class class)co 7362  ωcom 7807  1_oc1o 8410  2_oc2o 8411  ∈_𝑔cgoe 34014  ⊼_𝑔cgna 34015  ∀_𝑔cgol 34016  Fmlacfmla 34018

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9586

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-map 8774  df-goel 34021  df-gona 34022  df-goal 34023  df-sat 34024  df-fmla 34026

This theorem is referenced by:  satffunlem1lem2  34084