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Definition df-prt 33175
 Description: Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.)
Assertion
Ref Expression
df-prt (Prt 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
Distinct variable group:   𝑥,𝑦,𝐴

Detailed syntax breakdown of Definition df-prt
StepHypRef Expression
1 cA . . 3 class 𝐴
21wprt 33174 . 2 wff Prt 𝐴
3 vx . . . . . 6 setvar 𝑥
4 vy . . . . . 6 setvar 𝑦
53, 4weq 1861 . . . . 5 wff 𝑥 = 𝑦
63cv 1474 . . . . . . 7 class 𝑥
74cv 1474 . . . . . . 7 class 𝑦
86, 7cin 3539 . . . . . 6 class (𝑥𝑦)
9 c0 3874 . . . . . 6 class
108, 9wceq 1475 . . . . 5 wff (𝑥𝑦) = ∅
115, 10wo 382 . . . 4 wff (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)
1211, 4, 1wral 2896 . . 3 wff 𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)
1312, 3, 1wral 2896 . 2 wff 𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)
142, 13wb 195 1 wff (Prt 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
 Colors of variables: wff setvar class This definition is referenced by:  erprt  33176  prtlem14  33177
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