Definition df-nq 10599

Description: Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10808, 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 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-nq

Step	Hyp	Ref	Expression
1		cnq 10539	. 2 class Q
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1538	. . . . . 6 class 𝑥
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1538	. . . . . 6 class 𝑦
6		ceq 10538	. . . . . 6 class ~Q
7	3, 5, 6	wbr 5070	. . . . 5 wff 𝑥 ~Q 𝑦
8		c2nd 7803	. . . . . . . 8 class 2nd
9	5, 8	cfv 6418	. . . . . . 7 class (2nd ‘𝑦)
10	3, 8	cfv 6418	. . . . . . 7 class (2nd ‘𝑥)
11		clti 10534	. . . . . . 7 class <N
12	9, 10, 11	wbr 5070	. . . . . 6 wff (2nd ‘𝑦) <N (2nd ‘𝑥)
13	12	wn 3	. . . . 5 wff ¬ (2nd ‘𝑦) <N (2nd ‘𝑥)
14	7, 13	wi 4	. . . 4 wff (𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))
15		cnpi 10531	. . . . 5 class N
16	15, 15	cxp 5578	. . . 4 class (N × N)
17	14, 4, 16	wral 3063	. . 3 wff ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))
18	17, 2, 16	crab 3067	. 2 class {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}
19	1, 18	wceq 1539	1 wff Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}

This definition is referenced by:  nqex  10610  0nnq  10611  elpqn  10612  pinq  10614  nqereu  10616