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Definition df-nq 9694
Description: Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 9902, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 16-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-nq Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))}
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-nq
StepHypRef Expression
1 cnq 9634 . 2 class Q
2 vx . . . . . . 7 setvar 𝑥
32cv 1479 . . . . . 6 class 𝑥
4 vy . . . . . . 7 setvar 𝑦
54cv 1479 . . . . . 6 class 𝑦
6 ceq 9633 . . . . . 6 class ~Q
73, 5, 6wbr 4623 . . . . 5 wff 𝑥 ~Q 𝑦
8 c2nd 7127 . . . . . . . 8 class 2nd
95, 8cfv 5857 . . . . . . 7 class (2nd𝑦)
103, 8cfv 5857 . . . . . . 7 class (2nd𝑥)
11 clti 9629 . . . . . . 7 class <N
129, 10, 11wbr 4623 . . . . . 6 wff (2nd𝑦) <N (2nd𝑥)
1312wn 3 . . . . 5 wff ¬ (2nd𝑦) <N (2nd𝑥)
147, 13wi 4 . . . 4 wff (𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))
15 cnpi 9626 . . . . 5 class N
1615, 15cxp 5082 . . . 4 class (N × N)
1714, 4, 16wral 2908 . . 3 wff 𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))
1817, 2, 16crab 2912 . 2 class {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))}
191, 18wceq 1480 1 wff Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))}
Colors of variables: wff setvar class
This definition is referenced by:  nqex  9705  0nnq  9706  elpqn  9707  pinq  9709  nqereu  9711
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