Theorem funray 36128

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 3209	. . . . . 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 1136	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) → 𝑝 ∈ (𝔼‘𝑛))
3		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) → 𝑝 ∈ (𝔼‘𝑚))
4		axdimuniq 28840	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑚))) → 𝑛 = 𝑚)
5		fveq2 6858	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
6		rabeq 3420	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝔼‘𝑛) = (𝔼‘𝑚) → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})
7	5, 6	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})
8	7	eqeq2d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
9	8	anbi1d 631	. . . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → 𝑟 = 𝑠)
11	9, 10	biimtrdi 253	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → 𝑟 = 𝑠))
12	4, 11	syl 17	. . . . . . . . . . . . . 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 660	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 89	. . . . . . . . . . 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 596	. . . . . . . . . 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 406	. . . . . . . . 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 660	. . . . . . . 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 32	. . . . . . 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 3179	. . . . . 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 235	. . . . 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 1796	. . . 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 2733	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
24	23	anbi2d 630	. . . . . . 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 3157	. . . . . 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 2814	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑝 ∈ (𝔼‘𝑛) ↔ 𝑝 ∈ (𝔼‘𝑚)))
27	5	eleq2d 2814	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝑎 ∈ (𝔼‘𝑚)))
28	26, 27	3anbi12d 1439	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ↔ (𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎)))
29	7	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
30	28, 29	anbi12d 632	. . . . . . 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 3216	. . . . . 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 287	. . . . 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 2559	. . . 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 231	. . 3 ⊢ ∃*𝑟∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})
35	34	funoprab 7511	. 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 36126	. . 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 6537	. 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 231	1 ⊢ Fun Ray

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃*wmo 2531   ≠ wne 2925  ∃wrex 3053  {crab 3405  ⟨cop 4595   class class class wbr 5107  Fun wfun 6505  ‘cfv 6511  {coprab 7388  ℕcn 12186  𝔼cee 28815  OutsideOfcoutsideof 36107  Raycray 36123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  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 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-z 12530  df-uz 12794  df-fz 13469  df-ee 28818  df-ray 36126

This theorem is referenced by:  fvray  36129