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Definition df-ltxr 10367
Description: Define 'less than' on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. Note that in our postulates for complex numbers, < is primitive and not necessarily a relation on . (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
df-ltxr < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-ltxr
StepHypRef Expression
1 clt 10362 . 2 class <
2 vx . . . . . . 7 setvar 𝑥
32cv 1636 . . . . . 6 class 𝑥
4 cr 10223 . . . . . 6 class
53, 4wcel 2157 . . . . 5 wff 𝑥 ∈ ℝ
6 vy . . . . . . 7 setvar 𝑦
76cv 1636 . . . . . 6 class 𝑦
87, 4wcel 2157 . . . . 5 wff 𝑦 ∈ ℝ
9 cltrr 10228 . . . . . 6 class <
103, 7, 9wbr 4851 . . . . 5 wff 𝑥 < 𝑦
115, 8, 10w3a 1100 . . . 4 wff (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)
1211, 2, 6copab 4913 . . 3 class {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}
13 cmnf 10360 . . . . . . 7 class -∞
1413csn 4377 . . . . . 6 class {-∞}
154, 14cun 3774 . . . . 5 class (ℝ ∪ {-∞})
16 cpnf 10359 . . . . . 6 class +∞
1716csn 4377 . . . . 5 class {+∞}
1815, 17cxp 5316 . . . 4 class ((ℝ ∪ {-∞}) × {+∞})
1914, 4cxp 5316 . . . 4 class ({-∞} × ℝ)
2018, 19cun 3774 . . 3 class (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))
2112, 20cun 3774 . 2 class ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
221, 21wceq 1637 1 wff < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
Colors of variables: wff setvar class
This definition is referenced by:  ltrelxr  10387  ltxrlt  10396  ltxr  12168
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