Theorem brfs 33150

Description: Binary relation form of the general five segment predicate. (Contributed by Scott Fenton, 5-Oct-2013.)

Assertion

Ref	Expression
brfs	⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ FiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))))

Proof of Theorem brfs

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑝 𝑞 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4965	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 Colinear ⟨𝑏, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝑏, 𝑐⟩))
2		opeq1 4710	. . . 4 ⊢ (𝑎 = 𝐴 → ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ = ⟨𝐴, ⟨𝑏, 𝑐⟩⟩)
3	2	breq1d 4972	. . 3 ⊢ (𝑎 = 𝐴 → (⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
4		opeq1 4710	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑑⟩ = ⟨𝐴, 𝑑⟩)
5	4	breq1d 4972	. . . 4 ⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ↔ ⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩))
6	5	anbi1d 629	. . 3 ⊢ (𝑎 = 𝐴 → ((⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩)))
7	1, 3, 6	3anbi123d 1428	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩))))
8		opeq1 4710	. . . 4 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
9	8	breq2d 4974	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 Colinear ⟨𝑏, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝐵, 𝑐⟩))
10	8	opeq2d 4717	. . . 4 ⊢ (𝑏 = 𝐵 → ⟨𝐴, ⟨𝑏, 𝑐⟩⟩ = ⟨𝐴, ⟨𝐵, 𝑐⟩⟩)
11	10	breq1d 4972	. . 3 ⊢ (𝑏 = 𝐵 → (⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
12		opeq1 4710	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑑⟩ = ⟨𝐵, 𝑑⟩)
13	12	breq1d 4972	. . . 4 ⊢ (𝑏 = 𝐵 → (⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩ ↔ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩))
14	13	anbi2d 628	. . 3 ⊢ (𝑏 = 𝐵 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩)))
15	9, 11, 14	3anbi123d 1428	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝑐⟩ ∧ ⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩))))
16		opeq2 4711	. . . 4 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
17	16	breq2d 4974	. . 3 ⊢ (𝑐 = 𝐶 → (𝐴 Colinear ⟨𝐵, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝐵, 𝐶⟩))
18	16	opeq2d 4717	. . . 4 ⊢ (𝑐 = 𝐶 → ⟨𝐴, ⟨𝐵, 𝑐⟩⟩ = ⟨𝐴, ⟨𝐵, 𝐶⟩⟩)
19	18	breq1d 4972	. . 3 ⊢ (𝑐 = 𝐶 → (⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
20	17, 19	3anbi12d 1429	. 2 ⊢ (𝑐 = 𝐶 → ((𝐴 Colinear ⟨𝐵, 𝑐⟩ ∧ ⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩))))
21		opeq2 4711	. . . . 5 ⊢ (𝑑 = 𝐷 → ⟨𝐴, 𝑑⟩ = ⟨𝐴, 𝐷⟩)
22	21	breq1d 4972	. . . 4 ⊢ (𝑑 = 𝐷 → (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩))
23		opeq2 4711	. . . . 5 ⊢ (𝑑 = 𝐷 → ⟨𝐵, 𝑑⟩ = ⟨𝐵, 𝐷⟩)
24	23	breq1d 4972	. . . 4 ⊢ (𝑑 = 𝐷 → (⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩))
25	22, 24	anbi12d 630	. . 3 ⊢ (𝑑 = 𝐷 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)))
26	25	3anbi3d 1434	. 2 ⊢ (𝑑 = 𝐷 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩))))
27		opeq1 4710	. . . 4 ⊢ (𝑒 = 𝐸 → ⟨𝑒, ⟨𝑓, 𝑔⟩⟩ = ⟨𝐸, ⟨𝑓, 𝑔⟩⟩)
28	27	breq2d 4974	. . 3 ⊢ (𝑒 = 𝐸 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩))
29		opeq1 4710	. . . . 5 ⊢ (𝑒 = 𝐸 → ⟨𝑒, ℎ⟩ = ⟨𝐸, ℎ⟩)
30	29	breq2d 4974	. . . 4 ⊢ (𝑒 = 𝐸 → (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩))
31	30	anbi1d 629	. . 3 ⊢ (𝑒 = 𝐸 → ((⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)))
32	28, 31	3anbi23d 1431	. 2 ⊢ (𝑒 = 𝐸 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩))))
33		opeq1 4710	. . . . 5 ⊢ (𝑓 = 𝐹 → ⟨𝑓, 𝑔⟩ = ⟨𝐹, 𝑔⟩)
34	33	opeq2d 4717	. . . 4 ⊢ (𝑓 = 𝐹 → ⟨𝐸, ⟨𝑓, 𝑔⟩⟩ = ⟨𝐸, ⟨𝐹, 𝑔⟩⟩)
35	34	breq2d 4974	. . 3 ⊢ (𝑓 = 𝐹 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩))
36		opeq1 4710	. . . . 5 ⊢ (𝑓 = 𝐹 → ⟨𝑓, ℎ⟩ = ⟨𝐹, ℎ⟩)
37	36	breq2d 4974	. . . 4 ⊢ (𝑓 = 𝐹 → (⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩))
38	37	anbi2d 628	. . 3 ⊢ (𝑓 = 𝐹 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩)))
39	35, 38	3anbi23d 1431	. 2 ⊢ (𝑓 = 𝐹 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩))))
40		opeq2 4711	. . . . 5 ⊢ (𝑔 = 𝐺 → ⟨𝐹, 𝑔⟩ = ⟨𝐹, 𝐺⟩)
41	40	opeq2d 4717	. . . 4 ⊢ (𝑔 = 𝐺 → ⟨𝐸, ⟨𝐹, 𝑔⟩⟩ = ⟨𝐸, ⟨𝐹, 𝐺⟩⟩)
42	41	breq2d 4974	. . 3 ⊢ (𝑔 = 𝐺 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩))
43	42	3anbi2d 1433	. 2 ⊢ (𝑔 = 𝐺 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩))))
44		opeq2 4711	. . . . 5 ⊢ (ℎ = 𝐻 → ⟨𝐸, ℎ⟩ = ⟨𝐸, 𝐻⟩)
45	44	breq2d 4974	. . . 4 ⊢ (ℎ = 𝐻 → (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩))
46		opeq2 4711	. . . . 5 ⊢ (ℎ = 𝐻 → ⟨𝐹, ℎ⟩ = ⟨𝐹, 𝐻⟩)
47	46	breq2d 4974	. . . 4 ⊢ (ℎ = 𝐻 → (⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))
48	45, 47	anbi12d 630	. . 3 ⊢ (ℎ = 𝐻 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩)))
49	48	3anbi3d 1434	. 2 ⊢ (ℎ = 𝐻 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))))
50		fveq2 6538	. 2 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
51		df-fs 33113	. 2 ⊢ FiveSeg = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑒 ∈ (𝔼‘𝑛)∃𝑓 ∈ (𝔼‘𝑛)∃𝑔 ∈ (𝔼‘𝑛)∃ℎ ∈ (𝔼‘𝑛)(𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ (𝑎 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩)))}
52	7, 15, 20, 26, 32, 39, 43, 49, 50, 51	br8 32601	1 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ FiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ⟨cop 4478   class class class wbr 4962  ‘cfv 6225  ℕcn 11486  𝔼cee 26357  Cgrccgr 26359  Cgr3ccgr3 33107   Colinear ccolin 33108   FiveSeg cfs 33109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-iota 6189  df-fv 6233  df-fs 33113

This theorem is referenced by:  fscgr  33151  linecgr  33152