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Theorem netap 7253
Description: Negated equality on a set with decidable equality is a tight apartness. (Contributed by Jim Kingdon, 5-Feb-2025.)
Assertion
Ref Expression
netap (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} 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.
StepHypRef Expression
1 opabssxp 4701 . . 3 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ⊆ (𝐴 × 𝐴)
21a1i 9 . 2 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ⊆ (𝐴 × 𝐴))
3 neirr 2356 . . . . . 6 ¬ 𝑎𝑎
4 df-br 4005 . . . . . . 7 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎 ↔ ⟨𝑎, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)})
5 neeq1 2360 . . . . . . . . 9 (𝑢 = 𝑎 → (𝑢𝑣𝑎𝑣))
6 neeq2 2361 . . . . . . . . 9 (𝑣 = 𝑎 → (𝑎𝑣𝑎𝑎))
75, 6opelopab2 4271 . . . . . . . 8 ((𝑎𝐴𝑎𝐴) → (⟨𝑎, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ↔ 𝑎𝑎))
87anidms 397 . . . . . . 7 (𝑎𝐴 → (⟨𝑎, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ↔ 𝑎𝑎))
94, 8bitrid 192 . . . . . 6 (𝑎𝐴 → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎𝑎𝑎))
103, 9mtbiri 675 . . . . 5 (𝑎𝐴 → ¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎)
1110rgen 2530 . . . 4 𝑎𝐴 ¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎
1211a1i 9 . . 3 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → ∀𝑎𝐴 ¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎)
13 df-br 4005 . . . . . . . 8 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)})
14 neeq2 2361 . . . . . . . . 9 (𝑣 = 𝑏 → (𝑎𝑣𝑎𝑏))
155, 14opelopab2 4271 . . . . . . . 8 ((𝑎𝐴𝑏𝐴) → (⟨𝑎, 𝑏⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ↔ 𝑎𝑏))
1613, 15bitrid 192 . . . . . . 7 ((𝑎𝐴𝑏𝐴) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑎𝑏))
17 df-br 4005 . . . . . . . 8 (𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)})
18 neeq1 2360 . . . . . . . . . . 11 (𝑢 = 𝑏 → (𝑢𝑣𝑏𝑣))
19 neeq2 2361 . . . . . . . . . . 11 (𝑣 = 𝑎 → (𝑏𝑣𝑏𝑎))
2018, 19opelopab2 4271 . . . . . . . . . 10 ((𝑏𝐴𝑎𝐴) → (⟨𝑏, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ↔ 𝑏𝑎))
2120ancoms 268 . . . . . . . . 9 ((𝑎𝐴𝑏𝐴) → (⟨𝑏, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ↔ 𝑏𝑎))
22 necom 2431 . . . . . . . . 9 (𝑏𝑎𝑎𝑏)
2321, 22bitrdi 196 . . . . . . . 8 ((𝑎𝐴𝑏𝐴) → (⟨𝑏, 𝑎⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ↔ 𝑎𝑏))
2417, 23bitrid 192 . . . . . . 7 ((𝑎𝐴𝑏𝐴) → (𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎𝑎𝑏))
2516, 24bitr4d 191 . . . . . 6 ((𝑎𝐴𝑏𝐴) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎))
2625biimpd 144 . . . . 5 ((𝑎𝐴𝑏𝐴) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎))
2726rgen2 2563 . . . 4 𝑎𝐴𝑏𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎)
2827a1i 9 . . 3 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → ∀𝑎𝐴𝑏𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎))
2912, 28jca 306 . 2 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → (∀𝑎𝐴 ¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎)))
30163adant3 1017 . . . . . 6 ((𝑎𝐴𝑏𝐴𝑐𝐴) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑎𝑏))
3130adantl 277 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑎𝑏))
32 simpr 110 . . . . . . . . . . 11 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) ∧ 𝑎 = 𝑐) → 𝑎 = 𝑐)
33 simplr 528 . . . . . . . . . . 11 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) ∧ 𝑎 = 𝑐) → 𝑎𝑏)
3432, 33eqnetrrd 2373 . . . . . . . . . 10 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) ∧ 𝑎 = 𝑐) → 𝑐𝑏)
3534necomd 2433 . . . . . . . . 9 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) ∧ 𝑎 = 𝑐) → 𝑏𝑐)
3635olcd 734 . . . . . . . 8 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) ∧ 𝑎 = 𝑐) → (𝑎𝑐𝑏𝑐))
37 simpr 110 . . . . . . . . . 10 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) ∧ ¬ 𝑎 = 𝑐) → ¬ 𝑎 = 𝑐)
3837neqned 2354 . . . . . . . . 9 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) ∧ ¬ 𝑎 = 𝑐) → 𝑎𝑐)
3938orcd 733 . . . . . . . 8 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) ∧ ¬ 𝑎 = 𝑐) → (𝑎𝑐𝑏𝑐))
40 equequ2 1713 . . . . . . . . . . 11 (𝑦 = 𝑐 → (𝑎 = 𝑦𝑎 = 𝑐))
4140dcbid 838 . . . . . . . . . 10 (𝑦 = 𝑐 → (DECID 𝑎 = 𝑦DECID 𝑎 = 𝑐))
42 equequ1 1712 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑥 = 𝑦𝑎 = 𝑦))
4342dcbid 838 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (DECID 𝑥 = 𝑦DECID 𝑎 = 𝑦))
4443ralbidv 2477 . . . . . . . . . . 11 (𝑥 = 𝑎 → (∀𝑦𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑦𝐴 DECID 𝑎 = 𝑦))
45 simpll 527 . . . . . . . . . . 11 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
46 simplr1 1039 . . . . . . . . . . 11 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) → 𝑎𝐴)
4744, 45, 46rspcdva 2847 . . . . . . . . . 10 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) → ∀𝑦𝐴 DECID 𝑎 = 𝑦)
48 simplr3 1041 . . . . . . . . . 10 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) → 𝑐𝐴)
4941, 47, 48rspcdva 2847 . . . . . . . . 9 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) → DECID 𝑎 = 𝑐)
50 exmiddc 836 . . . . . . . . 9 (DECID 𝑎 = 𝑐 → (𝑎 = 𝑐 ∨ ¬ 𝑎 = 𝑐))
5149, 50syl 14 . . . . . . . 8 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) → (𝑎 = 𝑐 ∨ ¬ 𝑎 = 𝑐))
5236, 39, 51mpjaodan 798 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) → (𝑎𝑐𝑏𝑐))
53 df-br 4005 . . . . . . . . . 10 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐 ↔ ⟨𝑎, 𝑐⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)})
54 neeq2 2361 . . . . . . . . . . . 12 (𝑣 = 𝑐 → (𝑎𝑣𝑎𝑐))
555, 54opelopab2 4271 . . . . . . . . . . 11 ((𝑎𝐴𝑐𝐴) → (⟨𝑎, 𝑐⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ↔ 𝑎𝑐))
56553adant2 1016 . . . . . . . . . 10 ((𝑎𝐴𝑏𝐴𝑐𝐴) → (⟨𝑎, 𝑐⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ↔ 𝑎𝑐))
5753, 56bitrid 192 . . . . . . . . 9 ((𝑎𝐴𝑏𝐴𝑐𝐴) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐𝑎𝑐))
58 df-br 4005 . . . . . . . . . 10 (𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)})
59 neeq2 2361 . . . . . . . . . . . 12 (𝑣 = 𝑐 → (𝑏𝑣𝑏𝑐))
6018, 59opelopab2 4271 . . . . . . . . . . 11 ((𝑏𝐴𝑐𝐴) → (⟨𝑏, 𝑐⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ↔ 𝑏𝑐))
61603adant1 1015 . . . . . . . . . 10 ((𝑎𝐴𝑏𝐴𝑐𝐴) → (⟨𝑏, 𝑐⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ↔ 𝑏𝑐))
6258, 61bitrid 192 . . . . . . . . 9 ((𝑎𝐴𝑏𝐴𝑐𝐴) → (𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐𝑏𝑐))
6357, 62orbi12d 793 . . . . . . . 8 ((𝑎𝐴𝑏𝐴𝑐𝐴) → ((𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐) ↔ (𝑎𝑐𝑏𝑐)))
6463ad2antlr 489 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) → ((𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐) ↔ (𝑎𝑐𝑏𝑐)))
6552, 64mpbird 167 . . . . . 6 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) ∧ 𝑎𝑏) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐))
6665ex 115 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (𝑎𝑏 → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐)))
6731, 66sylbid 150 . . . 4 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏 → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐)))
6867ralrimivvva 2560 . . 3 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏 → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐)))
6916notbid 667 . . . . . . 7 ((𝑎𝐴𝑏𝐴) → (¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏 ↔ ¬ 𝑎𝑏))
70 df-ne 2348 . . . . . . . 8 (𝑎𝑏 ↔ ¬ 𝑎 = 𝑏)
7170notbii 668 . . . . . . 7 𝑎𝑏 ↔ ¬ ¬ 𝑎 = 𝑏)
7269, 71bitrdi 196 . . . . . 6 ((𝑎𝐴𝑏𝐴) → (¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏 ↔ ¬ ¬ 𝑎 = 𝑏))
7372adantl 277 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴)) → (¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏 ↔ ¬ ¬ 𝑎 = 𝑏))
74 equequ2 1713 . . . . . . . 8 (𝑦 = 𝑏 → (𝑎 = 𝑦𝑎 = 𝑏))
7574dcbid 838 . . . . . . 7 (𝑦 = 𝑏 → (DECID 𝑎 = 𝑦DECID 𝑎 = 𝑏))
76 simpl 109 . . . . . . . 8 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴)) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
77 simprl 529 . . . . . . . 8 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴)) → 𝑎𝐴)
7844, 76, 77rspcdva 2847 . . . . . . 7 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴)) → ∀𝑦𝐴 DECID 𝑎 = 𝑦)
79 simprr 531 . . . . . . 7 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴)) → 𝑏𝐴)
8075, 78, 79rspcdva 2847 . . . . . 6 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴)) → DECID 𝑎 = 𝑏)
81 notnotrdc 843 . . . . . 6 (DECID 𝑎 = 𝑏 → (¬ ¬ 𝑎 = 𝑏𝑎 = 𝑏))
8280, 81syl 14 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴)) → (¬ ¬ 𝑎 = 𝑏𝑎 = 𝑏))
8373, 82sylbid 150 . . . 4 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑎𝐴𝑏𝐴)) → (¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑎 = 𝑏))
8483ralrimivva 2559 . . 3 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → ∀𝑎𝐴𝑏𝐴𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑎 = 𝑏))
8568, 84jca 306 . 2 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → (∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏 → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐)) ∧ ∀𝑎𝐴𝑏𝐴𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑎 = 𝑏)))
86 dftap2 7250 . 2 ({⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} TAp 𝐴 ↔ ({⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} ⊆ (𝐴 × 𝐴) ∧ (∀𝑎𝐴 ¬ 𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑎)) ∧ (∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏 → (𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐𝑏{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑐)) ∧ ∀𝑎𝐴𝑏𝐴𝑎{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)}𝑏𝑎 = 𝑏))))
872, 29, 85, 86syl3anbrc 1181 1 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ 𝑢𝑣)} TAp 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  w3a 978  wcel 2148  wne 2347  wral 2455  wss 3130  cop 3596   class class class wbr 4004  {copab 4064   × cxp 4625   TAp wtap 7248
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 4122  ax-pow 4175  ax-pr 4210
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 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-pap 7247  df-tap 7249
This theorem is referenced by:  2onetap  7254  exmidapne  7259
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