Definition df-ltp 10401

Description: Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 10537, and is intended to be used only by the construction. From Proposition 9-3.2 of [Gleason] p. 122. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
df-ltp	⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-ltp

Step	Hyp	Ref	Expression
1		cltp 10279	. 2 class <P
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1532	. . . . . 6 class 𝑥
4		cnp 10275	. . . . . 6 class P
5	3, 4	wcel 2110	. . . . 5 wff 𝑥 ∈ P
6		vy	. . . . . . 7 setvar 𝑦
7	6	cv 1532	. . . . . 6 class 𝑦
8	7, 4	wcel 2110	. . . . 5 wff 𝑦 ∈ P
9	5, 8	wa 398	. . . 4 wff (𝑥 ∈ P ∧ 𝑦 ∈ P)
10	3, 7	wpss 3936	. . . 4 wff 𝑥 ⊊ 𝑦
11	9, 10	wa 398	. . 3 wff ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)
12	11, 2, 6	copab 5120	. 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
13	1, 12	wceq 1533	1 wff <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}

This definition is referenced by:  ltrelpr  10414  ltprord  10446