Theorem funray 36610

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 3237	. . . . . 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 29241	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑚))) → 𝑛 = 𝑚)
5		fveq2 6883	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
6		rabeq 3430	. . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
9	8	anbi1d 642	. . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . . . 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 672	. . . . . . . . . . . . 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 417	. . . . . . . . . . . 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 607	. . . . . . . . . 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 411	. . . . . . . . 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 672	. . . . . . . 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 3207	. . . . . 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 1826	. . . 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 2767	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
24	23	anbi2d 641	. . . . . . 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 3189	. . . . . 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 2849	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑝 ∈ (𝔼‘𝑛) ↔ 𝑝 ∈ (𝔼‘𝑚)))
27	5	eleq2d 2849	. . . . . . . . 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 2774	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
30	28, 29	anbi12d 643	. . . . . . 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 3244	. . . . . 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 2594	. . . 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 7534	. 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 36608	. . 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 6559	. 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

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2143  ∃*wmo 2565   ≠ wne 2958  ∃wrex 3089  {crab 3416  ⟨cop 4596   class class class wbr 5110  Fun wfun 6532  ‘cfv 6538  {coprab 7413  ℕcn 12234  𝔼cee 29215  OutsideOfcoutsideof 36589  Raycray 36605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-z 12593  df-uz 12864  df-fz 13537  df-ee 29218  df-ray 36608

This theorem is referenced by:  fvray  36611