Theorem funray 36728

Description: Show that the Ray relationship is a function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
funray	⊢ Fun Ray

Proof of Theorem funray

Dummy variables 𝑚 𝑎 𝑛 𝑝 𝑟 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reeanv 3236	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) ↔ (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})))
2		simp1 1154	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) → 𝑝 ∈ (𝔼‘𝑛))
3		simp1 1154	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) → 𝑝 ∈ (𝔼‘𝑚))
4		axdimuniq 29378	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑚))) → 𝑛 = 𝑚)
5		fveq2 6882	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
6		rabeq 3428	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝔼‘𝑛) = (𝔼‘𝑚) → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})
7	5, 6	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})
8	7	eqeq2d 2773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
9	8	anbi1d 643	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ (𝑟 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})))
10		eqtr3 2784	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → 𝑟 = 𝑠)
11	9, 10	biimtrdi 256	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → 𝑟 = 𝑠))
12	4, 11	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑚))) → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → 𝑟 = 𝑠))
13	12	an4s 673	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑝 ∈ (𝔼‘𝑚))) → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → 𝑟 = 𝑠))
14	13	ex 418	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑝 ∈ (𝔼‘𝑚)) → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → 𝑟 = 𝑠)))
15	14	com3l 90	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑝 ∈ (𝔼‘𝑚)) → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → 𝑟 = 𝑠)))
16	2, 3, 15	syl2an 608	. . . . . . . . . 10 ⊢ (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ (𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎)) → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → 𝑟 = 𝑠)))
17	16	imp 412	. . . . . . . . 9 ⊢ ((((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ (𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎)) ∧ (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → 𝑟 = 𝑠))
18	17	an4s 673	. . . . . . . 8 ⊢ ((((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → 𝑟 = 𝑠))
19	18	com12 33	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → 𝑟 = 𝑠))
20	19	rexlimivv 3206	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → 𝑟 = 𝑠)
21	1, 20	sylbir 238	. . . . 5 ⊢ ((∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → 𝑟 = 𝑠)
22	21	gen2 1829	. . . 4 ⊢ ∀𝑟∀𝑠((∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → 𝑟 = 𝑠)
23		eqeq1 2766	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
24	23	anbi2d 642	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})))
25	24	rexbidv 3188	. . . . . 6 ⊢ (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})))
26	5	eleq2d 2848	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑝 ∈ (𝔼‘𝑛) ↔ 𝑝 ∈ (𝔼‘𝑚)))
27	5	eleq2d 2848	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝑎 ∈ (𝔼‘𝑚)))
28	26, 27	3anbi12d 1465	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ↔ (𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎)))
29	7	eqeq2d 2773	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
30	28, 29	anbi12d 644	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})))
31	30	cbvrexvw 3243	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ∃𝑚 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
32	25, 31	bitrdi 290	. . . . 5 ⊢ (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ∃𝑚 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})))
33	32	mo4 2593	. . . 4 ⊢ (∃*𝑟∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ∀𝑟∀𝑠((∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → 𝑟 = 𝑠))
34	22, 33	mpbir 234	. . 3 ⊢ ∃*𝑟∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})
35	34	funoprab 7539	. 2 ⊢ Fun {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
36		df-ray 36726	. . 3 ⊢ Ray = {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
37	36	funeqi 6558	. 2 ⊢ (Fun Ray ↔ Fun {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})})
38	35, 37	mpbir 234	1 ⊢ Fun Ray

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃*wmo 2564   ≠ wne 2957  ∃wrex 3088  {crab 3414  ⟨cop 4593   class class class wbr 5107  Fun wfun 6531  ‘cfv 6537  {coprab 7418  ℕcn 12261  𝔼cee 29352  OutsideOfcoutsideof 36707  Raycray 36723

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-z 12620  df-uz 12892  df-fz 13566  df-ee 29355  df-ray 36726

This theorem is used by:  fvray  36729