Theorem aptap 8829

Description: Complex apartness (as defined at df-ap 8761) is a tight apartness (as defined at df-tap 7468). (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 2238	. . . . . . . . . 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 2557	. . . . . . 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 2557	. . . . . 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 2238	. . . . . . . . . 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 2557	. . . . . . 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 2557	. . . . . 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 6359	. . . . 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 8761	. . . . 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 2326	. . . 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 4855	. . . . . . . . . 10 ⊢ Rel #
15		simp-5l 545	. . . . . . . . . 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 6343	. . . . . . . . . 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 539	. . . . . . . . . . 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 546	. . . . . . . . . . . . 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 8207	. . . . . . . . . . . 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 8126	. . . . . . . . . . . . . 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 544	. . . . . . . . . . . . . 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 8207	. . . . . . . . . . . . 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 8199	. . . . . . . . . . . 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 8198	. . . . . . . . . . 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 2308	. . . . . . . . . 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 540	. . . . . . . . . . 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 536	. . . . . . . . . . . . 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 8207	. . . . . . . . . . . 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 529	. . . . . . . . . . . . . 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 8207	. . . . . . . . . . . . 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 8199	. . . . . . . . . . . 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 8198	. . . . . . . . . . 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 2308	. . . . . . . . . 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 6331	. . . . . . . . 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 2653	. . . . . . 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 2650	. . . . . 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 2650	. . . . 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 2650	. . . 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 3231	. 2 ⊢ # ⊆ (ℂ × ℂ)
45		apirr 8784	. . . 4 ⊢ (𝑥 ∈ ℂ → ¬ 𝑥 # 𝑥)
46	45	rgen 2585	. . 3 ⊢ ∀𝑥 ∈ ℂ ¬ 𝑥 # 𝑥
47		apsym 8785	. . . . 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 2618	. . 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 8786	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 # 𝑦 → (𝑥 # 𝑧 ∨ 𝑦 # 𝑧)))
52	51	rgen3 2619	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ (𝑥 # 𝑦 → (𝑥 # 𝑧 ∨ 𝑦 # 𝑧))
53		apti 8801	. . . . 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 2618	. . 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 7469	. 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 1205	1 ⊢ # TAp ℂ

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 715   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511   ⊆ wss 3200  ⟨cop 3672   class class class wbr 4088  {copab 4149   × cxp 4723  Rel wrel 4730  ‘cfv 5326  (class class class)co 6017  1^st c1st 6300  2^nd c2nd 6301   TAp wtap 7467  ℂcc 8029  ℝcr 8030  ici 8033   + caddc 8034   · cmul 8036   #_ℝ creap 8753   # cap 8760

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pap 7466  df-tap 7468  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761

This theorem is referenced by: (None)