Theorem aptap 8694

Description: Complex apartness (as defined at df-ap 8626) is a tight apartness (as defined at df-tap 7333). (Contributed by Jim Kingdon, 16-Feb-2025.)

Assertion

Ref	Expression
aptap	⊢ # TAp ℂ

Proof of Theorem aptap

Dummy variables 𝑞 𝑝 𝑟 𝑠 𝑡 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2203	. . . . . . . . . 10 ⊢ (𝑢 = (1st ‘𝑡) → (𝑢 = (𝑝 + (i · 𝑞)) ↔ (1st ‘𝑡) = (𝑝 + (i · 𝑞))))
2	1	anbi1d 465	. . . . . . . . 9 ⊢ (𝑢 = (1st ‘𝑡) → ((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ↔ ((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠)))))
3	2	anbi1d 465	. . . . . . . 8 ⊢ (𝑢 = (1st ‘𝑡) → (((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) ↔ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))))
4	3	2rexbidv 2522	. . . . . . 7 ⊢ (𝑢 = (1st ‘𝑡) → (∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) ↔ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))))
5	4	2rexbidv 2522	. . . . . 6 ⊢ (𝑢 = (1st ‘𝑡) → (∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) ↔ ∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))))
6		eqeq1 2203	. . . . . . . . . 10 ⊢ (𝑣 = (2nd ‘𝑡) → (𝑣 = (𝑟 + (i · 𝑠)) ↔ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))))
7	6	anbi2d 464	. . . . . . . . 9 ⊢ (𝑣 = (2nd ‘𝑡) → (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ↔ ((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠)))))
8	7	anbi1d 465	. . . . . . . 8 ⊢ (𝑣 = (2nd ‘𝑡) → ((((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) ↔ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))))
9	8	2rexbidv 2522	. . . . . . 7 ⊢ (𝑣 = (2nd ‘𝑡) → (∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) ↔ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))))
10	9	2rexbidv 2522	. . . . . 6 ⊢ (𝑣 = (2nd ‘𝑡) → (∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) ↔ ∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))))
11	5, 10	elopabi 6262	. . . . 5 ⊢ (𝑡 ∈ {⟨𝑢, 𝑣⟩ ∣ ∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))} → ∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)))
12		df-ap 8626	. . . . 5 ⊢ # = {⟨𝑢, 𝑣⟩ ∣ ∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))}
13	11, 12	eleq2s 2291	. . . 4 ⊢ (𝑡 ∈ # → ∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)))
14	12	relopabi 4792	. . . . . . . . . 10 ⊢ Rel #
15		simp-5l 543	. . . . . . . . . 10 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑡 ∈ # )
16		1st2nd 6248	. . . . . . . . . 10 ⊢ ((Rel # ∧ 𝑡 ∈ # ) → 𝑡 = ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩)
17	14, 15, 16	sylancr 414	. . . . . . . . 9 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑡 = ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩)
18		simprll 537	. . . . . . . . . . 11 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (1st ‘𝑡) = (𝑝 + (i · 𝑞)))
19		simp-5r 544	. . . . . . . . . . . . 13 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑝 ∈ ℝ)
20	19	recnd 8072	. . . . . . . . . . . 12 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑝 ∈ ℂ)
21		ax-icn 7991	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
22	21	a1i 9	. . . . . . . . . . . . 13 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → i ∈ ℂ)
23		simp-4r 542	. . . . . . . . . . . . . 14 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑞 ∈ ℝ)
24	23	recnd 8072	. . . . . . . . . . . . 13 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑞 ∈ ℂ)
25	22, 24	mulcld 8064	. . . . . . . . . . . 12 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (i · 𝑞) ∈ ℂ)
26	20, 25	addcld 8063	. . . . . . . . . . 11 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (𝑝 + (i · 𝑞)) ∈ ℂ)
27	18, 26	eqeltrd 2273	. . . . . . . . . 10 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (1st ‘𝑡) ∈ ℂ)
28		simprlr 538	. . . . . . . . . . 11 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (2nd ‘𝑡) = (𝑟 + (i · 𝑠)))
29		simpllr 534	. . . . . . . . . . . . 13 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑟 ∈ ℝ)
30	29	recnd 8072	. . . . . . . . . . . 12 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑟 ∈ ℂ)
31		simplr 528	. . . . . . . . . . . . . 14 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑠 ∈ ℝ)
32	31	recnd 8072	. . . . . . . . . . . . 13 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑠 ∈ ℂ)
33	22, 32	mulcld 8064	. . . . . . . . . . . 12 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (i · 𝑠) ∈ ℂ)
34	30, 33	addcld 8063	. . . . . . . . . . 11 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (𝑟 + (i · 𝑠)) ∈ ℂ)
35	28, 34	eqeltrd 2273	. . . . . . . . . 10 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (2nd ‘𝑡) ∈ ℂ)
36	27, 35	jca 306	. . . . . . . . 9 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → ((1st ‘𝑡) ∈ ℂ ∧ (2nd ‘𝑡) ∈ ℂ))
37		elxp6 6236	. . . . . . . . 9 ⊢ (𝑡 ∈ (ℂ × ℂ) ↔ (𝑡 = ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∧ ((1st ‘𝑡) ∈ ℂ ∧ (2nd ‘𝑡) ∈ ℂ)))
38	17, 36, 37	sylanbrc 417	. . . . . . . 8 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑡 ∈ (ℂ × ℂ))
39	38	rexlimdva2 2617	. . . . . . 7 ⊢ ((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) → (∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) → 𝑡 ∈ (ℂ × ℂ)))
40	39	rexlimdva 2614	. . . . . 6 ⊢ (((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) → (∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) → 𝑡 ∈ (ℂ × ℂ)))
41	40	rexlimdva 2614	. . . . 5 ⊢ ((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) → (∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) → 𝑡 ∈ (ℂ × ℂ)))
42	41	rexlimdva 2614	. . . 4 ⊢ (𝑡 ∈ # → (∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) → 𝑡 ∈ (ℂ × ℂ)))
43	13, 42	mpd 13	. . 3 ⊢ (𝑡 ∈ # → 𝑡 ∈ (ℂ × ℂ))
44	43	ssriv 3188	. 2 ⊢ # ⊆ (ℂ × ℂ)
45		apirr 8649	. . . 4 ⊢ (𝑥 ∈ ℂ → ¬ 𝑥 # 𝑥)
46	45	rgen 2550	. . 3 ⊢ ∀𝑥 ∈ ℂ ¬ 𝑥 # 𝑥
47		apsym 8650	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 # 𝑦 ↔ 𝑦 # 𝑥))
48	47	biimpd 144	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 # 𝑦 → 𝑦 # 𝑥))
49	48	rgen2 2583	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 # 𝑦 → 𝑦 # 𝑥)
50	46, 49	pm3.2i 272	. 2 ⊢ (∀𝑥 ∈ ℂ ¬ 𝑥 # 𝑥 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 # 𝑦 → 𝑦 # 𝑥))
51		apcotr 8651	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 # 𝑦 → (𝑥 # 𝑧 ∨ 𝑦 # 𝑧)))
52	51	rgen3 2584	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ (𝑥 # 𝑦 → (𝑥 # 𝑧 ∨ 𝑦 # 𝑧))
53		apti 8666	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 = 𝑦 ↔ ¬ 𝑥 # 𝑦))
54	53	biimprd 158	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (¬ 𝑥 # 𝑦 → 𝑥 = 𝑦))
55	54	rgen2 2583	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (¬ 𝑥 # 𝑦 → 𝑥 = 𝑦)
56	52, 55	pm3.2i 272	. 2 ⊢ (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ (𝑥 # 𝑦 → (𝑥 # 𝑧 ∨ 𝑦 # 𝑧)) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (¬ 𝑥 # 𝑦 → 𝑥 = 𝑦))
57		dftap2 7334	. 2 ⊢ ( # TAp ℂ ↔ ( # ⊆ (ℂ × ℂ) ∧ (∀𝑥 ∈ ℂ ¬ 𝑥 # 𝑥 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 # 𝑦 → 𝑦 # 𝑥)) ∧ (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ (𝑥 # 𝑦 → (𝑥 # 𝑧 ∨ 𝑦 # 𝑧)) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (¬ 𝑥 # 𝑦 → 𝑥 = 𝑦))))
58	44, 50, 56, 57	mpbir3an 1181	1 ⊢ # TAp ℂ

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476   ⊆ wss 3157  ⟨cop 3626   class class class wbr 4034  {copab 4094   × cxp 4662  Rel wrel 4669  ‘cfv 5259  (class class class)co 5925  1^st c1st 6205  2^nd c2nd 6206   TAp wtap 7332  ℂcc 7894  ℝcr 7895  ici 7898   + caddc 7899   · cmul 7901   #_ℝ creap 8618   # cap 8625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fo 5265  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pap 7331  df-tap 7333  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626

This theorem is referenced by: (None)