Theorem brfs 34381

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 5077	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 Colinear ⟨𝑏, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝑏, 𝑐⟩))
2		opeq1 4804	. . . 4 ⊢ (𝑎 = 𝐴 → ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ = ⟨𝐴, ⟨𝑏, 𝑐⟩⟩)
3	2	breq1d 5084	. . 3 ⊢ (𝑎 = 𝐴 → (⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
4		opeq1 4804	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑑⟩ = ⟨𝐴, 𝑑⟩)
5	4	breq1d 5084	. . . 4 ⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ↔ ⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩))
6	5	anbi1d 630	. . 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 1435	. 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 4804	. . . 4 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
9	8	breq2d 5086	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 Colinear ⟨𝑏, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝐵, 𝑐⟩))
10	8	opeq2d 4811	. . . 4 ⊢ (𝑏 = 𝐵 → ⟨𝐴, ⟨𝑏, 𝑐⟩⟩ = ⟨𝐴, ⟨𝐵, 𝑐⟩⟩)
11	10	breq1d 5084	. . 3 ⊢ (𝑏 = 𝐵 → (⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
12		opeq1 4804	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑑⟩ = ⟨𝐵, 𝑑⟩)
13	12	breq1d 5084	. . . 4 ⊢ (𝑏 = 𝐵 → (⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩ ↔ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩))
14	13	anbi2d 629	. . 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 1435	. 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 4805	. . . 4 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
17	16	breq2d 5086	. . 3 ⊢ (𝑐 = 𝐶 → (𝐴 Colinear ⟨𝐵, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝐵, 𝐶⟩))
18	16	opeq2d 4811	. . . 4 ⊢ (𝑐 = 𝐶 → ⟨𝐴, ⟨𝐵, 𝑐⟩⟩ = ⟨𝐴, ⟨𝐵, 𝐶⟩⟩)
19	18	breq1d 5084	. . 3 ⊢ (𝑐 = 𝐶 → (⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
20	17, 19	3anbi12d 1436	. 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 4805	. . . . 5 ⊢ (𝑑 = 𝐷 → ⟨𝐴, 𝑑⟩ = ⟨𝐴, 𝐷⟩)
22	21	breq1d 5084	. . . 4 ⊢ (𝑑 = 𝐷 → (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩))
23		opeq2 4805	. . . . 5 ⊢ (𝑑 = 𝐷 → ⟨𝐵, 𝑑⟩ = ⟨𝐵, 𝐷⟩)
24	23	breq1d 5084	. . . 4 ⊢ (𝑑 = 𝐷 → (⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩))
25	22, 24	anbi12d 631	. . 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 1441	. 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 4804	. . . 4 ⊢ (𝑒 = 𝐸 → ⟨𝑒, ⟨𝑓, 𝑔⟩⟩ = ⟨𝐸, ⟨𝑓, 𝑔⟩⟩)
28	27	breq2d 5086	. . 3 ⊢ (𝑒 = 𝐸 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩))
29		opeq1 4804	. . . . 5 ⊢ (𝑒 = 𝐸 → ⟨𝑒, ℎ⟩ = ⟨𝐸, ℎ⟩)
30	29	breq2d 5086	. . . 4 ⊢ (𝑒 = 𝐸 → (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩))
31	30	anbi1d 630	. . 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 1438	. 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 4804	. . . . 5 ⊢ (𝑓 = 𝐹 → ⟨𝑓, 𝑔⟩ = ⟨𝐹, 𝑔⟩)
34	33	opeq2d 4811	. . . 4 ⊢ (𝑓 = 𝐹 → ⟨𝐸, ⟨𝑓, 𝑔⟩⟩ = ⟨𝐸, ⟨𝐹, 𝑔⟩⟩)
35	34	breq2d 5086	. . 3 ⊢ (𝑓 = 𝐹 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩))
36		opeq1 4804	. . . . 5 ⊢ (𝑓 = 𝐹 → ⟨𝑓, ℎ⟩ = ⟨𝐹, ℎ⟩)
37	36	breq2d 5086	. . . 4 ⊢ (𝑓 = 𝐹 → (⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩))
38	37	anbi2d 629	. . 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 1438	. 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 4805	. . . . 5 ⊢ (𝑔 = 𝐺 → ⟨𝐹, 𝑔⟩ = ⟨𝐹, 𝐺⟩)
41	40	opeq2d 4811	. . . 4 ⊢ (𝑔 = 𝐺 → ⟨𝐸, ⟨𝐹, 𝑔⟩⟩ = ⟨𝐸, ⟨𝐹, 𝐺⟩⟩)
42	41	breq2d 5086	. . 3 ⊢ (𝑔 = 𝐺 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩))
43	42	3anbi2d 1440	. 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 4805	. . . . 5 ⊢ (ℎ = 𝐻 → ⟨𝐸, ℎ⟩ = ⟨𝐸, 𝐻⟩)
45	44	breq2d 5086	. . . 4 ⊢ (ℎ = 𝐻 → (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩))
46		opeq2 4805	. . . . 5 ⊢ (ℎ = 𝐻 → ⟨𝐹, ℎ⟩ = ⟨𝐹, 𝐻⟩)
47	46	breq2d 5086	. . . 4 ⊢ (ℎ = 𝐻 → (⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))
48	45, 47	anbi12d 631	. . 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 1441	. 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 6774	. 2 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
51		df-fs 34344	. 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 33723	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ⟨cop 4567   class class class wbr 5074  ‘cfv 6433  ℕcn 11973  𝔼cee 27256  Cgrccgr 27258  Cgr3ccgr3 34338   Colinear ccolin 34339   FiveSeg cfs 34340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-iota 6391  df-fv 6441  df-fs 34344

This theorem is referenced by:  fscgr  34382  linecgr  34383