Theorem netap 7252

Description: Negated equality on a set with decidable equality is a tight apartness. (Contributed by Jim Kingdon, 5-Feb-2025.)

Assertion

Ref	Expression
netap	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} TAp 𝐴)

Distinct variable groups:   𝑢,𝐴,𝑣   𝑥,𝐴,𝑦

Proof of Theorem netap

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opabssxp 4700	. . 3 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ⊆ (𝐴 × 𝐴)
2	1	a1i 9	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ⊆ (𝐴 × 𝐴))
3		neirr 2356	. . . . . 6 ⊢ ¬ 𝑎 ≠ 𝑎
4		df-br 4004	. . . . . . 7 ⊢ (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎 ↔ ⟨𝑎, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)})
5		neeq1 2360	. . . . . . . . 9 ⊢ (𝑢 = 𝑎 → (𝑢 ≠ 𝑣 ↔ 𝑎 ≠ 𝑣))
6		neeq2 2361	. . . . . . . . 9 ⊢ (𝑣 = 𝑎 → (𝑎 ≠ 𝑣 ↔ 𝑎 ≠ 𝑎))
7	5, 6	opelopab2 4270	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴) → (⟨𝑎, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ↔ 𝑎 ≠ 𝑎))
8	7	anidms 397	. . . . . . 7 ⊢ (𝑎 ∈ 𝐴 → (⟨𝑎, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ↔ 𝑎 ≠ 𝑎))
9	4, 8	bitrid 192	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎 ↔ 𝑎 ≠ 𝑎))
10	3, 9	mtbiri 675	. . . . 5 ⊢ (𝑎 ∈ 𝐴 → ¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎)
11	10	rgen 2530	. . . 4 ⊢ ∀𝑎 ∈ 𝐴 ¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎
12	11	a1i 9	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑎 ∈ 𝐴 ¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎)
13		df-br 4004	. . . . . . . 8 ⊢ (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)})
14		neeq2 2361	. . . . . . . . 9 ⊢ (𝑣 = 𝑏 → (𝑎 ≠ 𝑣 ↔ 𝑎 ≠ 𝑏))
15	5, 14	opelopab2 4270	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (⟨𝑎, 𝑏⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ↔ 𝑎 ≠ 𝑏))
16	13, 15	bitrid 192	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 ↔ 𝑎 ≠ 𝑏))
17		df-br 4004	. . . . . . . 8 ⊢ (𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)})
18		neeq1 2360	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑏 → (𝑢 ≠ 𝑣 ↔ 𝑏 ≠ 𝑣))
19		neeq2 2361	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑎 → (𝑏 ≠ 𝑣 ↔ 𝑏 ≠ 𝑎))
20	18, 19	opelopab2 4270	. . . . . . . . . 10 ⊢ ((𝑏 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴) → (⟨𝑏, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ↔ 𝑏 ≠ 𝑎))
21	20	ancoms 268	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (⟨𝑏, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ↔ 𝑏 ≠ 𝑎))
22		necom 2431	. . . . . . . . 9 ⊢ (𝑏 ≠ 𝑎 ↔ 𝑎 ≠ 𝑏)
23	21, 22	bitrdi 196	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (⟨𝑏, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ↔ 𝑎 ≠ 𝑏))
24	17, 23	bitrid 192	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎 ↔ 𝑎 ≠ 𝑏))
25	16, 24	bitr4d 191	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 ↔ 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎))
26	25	biimpd 144	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎))
27	26	rgen2 2563	. . . 4 ⊢ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎)
28	27	a1i 9	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎))
29	12, 28	jca 306	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → (∀𝑎 ∈ 𝐴 ¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎)))
30	16	3adant3 1017	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 ↔ 𝑎 ≠ 𝑏))
31	30	adantl 277	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 ↔ 𝑎 ≠ 𝑏))
32		simpr 110	. . . . . . . . . . 11 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) ∧ 𝑎 = 𝑐) → 𝑎 = 𝑐)
33		simplr 528	. . . . . . . . . . 11 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) ∧ 𝑎 = 𝑐) → 𝑎 ≠ 𝑏)
34	32, 33	eqnetrrd 2373	. . . . . . . . . 10 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) ∧ 𝑎 = 𝑐) → 𝑐 ≠ 𝑏)
35	34	necomd 2433	. . . . . . . . 9 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) ∧ 𝑎 = 𝑐) → 𝑏 ≠ 𝑐)
36	35	olcd 734	. . . . . . . 8 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) ∧ 𝑎 = 𝑐) → (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑐))
37		simpr 110	. . . . . . . . . 10 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) ∧ ¬ 𝑎 = 𝑐) → ¬ 𝑎 = 𝑐)
38	37	neqned 2354	. . . . . . . . 9 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) ∧ ¬ 𝑎 = 𝑐) → 𝑎 ≠ 𝑐)
39	38	orcd 733	. . . . . . . 8 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) ∧ ¬ 𝑎 = 𝑐) → (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑐))
40		equequ2 1713	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑐 → (𝑎 = 𝑦 ↔ 𝑎 = 𝑐))
41	40	dcbid 838	. . . . . . . . . 10 ⊢ (𝑦 = 𝑐 → (DECID 𝑎 = 𝑦 ↔ DECID 𝑎 = 𝑐))
42		equequ1 1712	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝑦 ↔ 𝑎 = 𝑦))
43	42	dcbid 838	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑎 = 𝑦))
44	43	ralbidv 2477	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐴 DECID 𝑎 = 𝑦))
45		simpll 527	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
46		simplr1 1039	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) → 𝑎 ∈ 𝐴)
47	44, 45, 46	rspcdva 2846	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) → ∀𝑦 ∈ 𝐴 DECID 𝑎 = 𝑦)
48		simplr3 1041	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) → 𝑐 ∈ 𝐴)
49	41, 47, 48	rspcdva 2846	. . . . . . . . 9 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) → DECID 𝑎 = 𝑐)
50		exmiddc 836	. . . . . . . . 9 ⊢ (DECID 𝑎 = 𝑐 → (𝑎 = 𝑐 ∨ ¬ 𝑎 = 𝑐))
51	49, 50	syl 14	. . . . . . . 8 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) → (𝑎 = 𝑐 ∨ ¬ 𝑎 = 𝑐))
52	36, 39, 51	mpjaodan 798	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) → (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑐))
53		df-br 4004	. . . . . . . . . 10 ⊢ (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐 ↔ ⟨𝑎, 𝑐⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)})
54		neeq2 2361	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑐 → (𝑎 ≠ 𝑣 ↔ 𝑎 ≠ 𝑐))
55	5, 54	opelopab2 4270	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (⟨𝑎, 𝑐⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ↔ 𝑎 ≠ 𝑐))
56	55	3adant2 1016	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (⟨𝑎, 𝑐⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ↔ 𝑎 ≠ 𝑐))
57	53, 56	bitrid 192	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐 ↔ 𝑎 ≠ 𝑐))
58		df-br 4004	. . . . . . . . . 10 ⊢ (𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)})
59		neeq2 2361	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑐 → (𝑏 ≠ 𝑣 ↔ 𝑏 ≠ 𝑐))
60	18, 59	opelopab2 4270	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (⟨𝑏, 𝑐⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ↔ 𝑏 ≠ 𝑐))
61	60	3adant1 1015	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (⟨𝑏, 𝑐⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ↔ 𝑏 ≠ 𝑐))
62	58, 61	bitrid 192	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐 ↔ 𝑏 ≠ 𝑐))
63	57, 62	orbi12d 793	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ((𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐 ∨ 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐) ↔ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑐)))
64	63	ad2antlr 489	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) → ((𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐 ∨ 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐) ↔ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑐)))
65	52, 64	mpbird 167	. . . . . 6 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) ∧ 𝑎 ≠ 𝑏) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐 ∨ 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐))
66	65	ex 115	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑎 ≠ 𝑏 → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐 ∨ 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐)))
67	31, 66	sylbid 150	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐 ∨ 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐)))
68	67	ralrimivvva 2560	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐 ∨ 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐)))
69	16	notbid 667	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 ↔ ¬ 𝑎 ≠ 𝑏))
70		df-ne 2348	. . . . . . . 8 ⊢ (𝑎 ≠ 𝑏 ↔ ¬ 𝑎 = 𝑏)
71	70	notbii 668	. . . . . . 7 ⊢ (¬ 𝑎 ≠ 𝑏 ↔ ¬ ¬ 𝑎 = 𝑏)
72	69, 71	bitrdi 196	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 ↔ ¬ ¬ 𝑎 = 𝑏))
73	72	adantl 277	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 ↔ ¬ ¬ 𝑎 = 𝑏))
74		equequ2 1713	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (𝑎 = 𝑦 ↔ 𝑎 = 𝑏))
75	74	dcbid 838	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (DECID 𝑎 = 𝑦 ↔ DECID 𝑎 = 𝑏))
76		simpl 109	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
77		simprl 529	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑎 ∈ 𝐴)
78	44, 76, 77	rspcdva 2846	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ∀𝑦 ∈ 𝐴 DECID 𝑎 = 𝑦)
79		simprr 531	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏 ∈ 𝐴)
80	75, 78, 79	rspcdva 2846	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → DECID 𝑎 = 𝑏)
81		notnotrdc 843	. . . . . 6 ⊢ (DECID 𝑎 = 𝑏 → (¬ ¬ 𝑎 = 𝑏 → 𝑎 = 𝑏))
82	80, 81	syl 14	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (¬ ¬ 𝑎 = 𝑏 → 𝑎 = 𝑏))
83	73, 82	sylbid 150	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → 𝑎 = 𝑏))
84	83	ralrimivva 2559	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → 𝑎 = 𝑏))
85	68, 84	jca 306	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐 ∨ 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐)) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → 𝑎 = 𝑏)))
86		dftap2 7249	. 2 ⊢ ({⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} TAp 𝐴 ↔ ({⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ⊆ (𝐴 × 𝐴) ∧ (∀𝑎 ∈ 𝐴 ¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑎)) ∧ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐 ∨ 𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑐)) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}𝑏 → 𝑎 = 𝑏))))
87	2, 29, 85, 86	syl3anbrc 1181	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} TAp 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834   ∧ w3a 978   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455   ⊆ wss 3129  ⟨cop 3595   class class class wbr 4003  {copab 4063   × cxp 4624   TAp wtap 7247

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-pap 7246  df-tap 7248

This theorem is referenced by:  2onetap  7253  exmidapne  7258