Theorem funray 36141

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 3229	. . . . . 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 1137	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) → 𝑝 ∈ (𝔼‘𝑛))
3		simp1 1137	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) → 𝑝 ∈ (𝔼‘𝑚))
4		axdimuniq 28928	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑚))) → 𝑛 = 𝑚)
5		fveq2 6906	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
6		rabeq 3451	. . . . . . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . 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 3201	. . . . . 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 2741	. . . . . . . 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 3179	. . . . . 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 2827	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑝 ∈ (𝔼‘𝑛) ↔ 𝑝 ∈ (𝔼‘𝑚)))
27	5	eleq2d 2827	. . . . . . . . 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 2748	. . . . . . . 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 3238	. . . . . 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 2566	. . . 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 7555	. 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 36139	. . 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 6587	. 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 1087  ∀wal 1538   = wceq 1540   ∈ wcel 2108  ∃*wmo 2538   ≠ wne 2940  ∃wrex 3070  {crab 3436  ⟨cop 4632   class class class wbr 5143  Fun wfun 6555  ‘cfv 6561  {coprab 7432  ℕcn 12266  𝔼cee 28903  OutsideOfcoutsideof 36120  Raycray 36136

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-z 12614  df-uz 12879  df-fz 13548  df-ee 28906  df-ray 36139

This theorem is referenced by:  fvray  36142