Theorem ltrelxr 10691

Description: "Less than" is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
ltrelxr	⊢ < ⊆ (ℝ* × ℝ*)

Proof of Theorem ltrelxr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ltxr 10669	. 2 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2		df-3an 1086	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦))
3	2	opabbii 5097	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)}
4		opabssxp 5607	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ × ℝ)
5	3, 4	eqsstri 3949	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ × ℝ)
6		rexpssxrxp 10675	. . . 4 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
7	5, 6	sstri 3924	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ* × ℝ*)
8		ressxr 10674	. . . . . 6 ⊢ ℝ ⊆ ℝ*
9		snsspr2 4708	. . . . . . 7 ⊢ {-∞} ⊆ {+∞, -∞}
10		ssun2 4100	. . . . . . . 8 ⊢ {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞})
11		df-xr 10668	. . . . . . . 8 ⊢ ℝ* = (ℝ ∪ {+∞, -∞})
12	10, 11	sseqtrri 3952	. . . . . . 7 ⊢ {+∞, -∞} ⊆ ℝ*
13	9, 12	sstri 3924	. . . . . 6 ⊢ {-∞} ⊆ ℝ*
14	8, 13	unssi 4112	. . . . 5 ⊢ (ℝ ∪ {-∞}) ⊆ ℝ*
15		snsspr1 4707	. . . . . 6 ⊢ {+∞} ⊆ {+∞, -∞}
16	15, 12	sstri 3924	. . . . 5 ⊢ {+∞} ⊆ ℝ*
17		xpss12 5534	. . . . 5 ⊢ (((ℝ ∪ {-∞}) ⊆ ℝ* ∧ {+∞} ⊆ ℝ*) → ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*))
18	14, 16, 17	mp2an 691	. . . 4 ⊢ ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*)
19		xpss12 5534	. . . . 5 ⊢ (({-∞} ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ({-∞} × ℝ) ⊆ (ℝ* × ℝ*))
20	13, 8, 19	mp2an 691	. . . 4 ⊢ ({-∞} × ℝ) ⊆ (ℝ* × ℝ*)
21	18, 20	unssi 4112	. . 3 ⊢ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ⊆ (ℝ* × ℝ*)
22	7, 21	unssi 4112	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))) ⊆ (ℝ* × ℝ*)
23	1, 22	eqsstri 3949	1 ⊢ < ⊆ (ℝ* × ℝ*)

Syntax hints:   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111   ∪ cun 3879   ⊆ wss 3881  {csn 4525  {cpr 4527   class class class wbr 5030  {copab 5092   × cxp 5517  ℝcr 10525   <_ℝ cltrr 10530  +∞cpnf 10661  -∞cmnf 10662  ℝ^*cxr 10663   < clt 10664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-pr 4528  df-opab 5093  df-xp 5525  df-xr 10668  df-ltxr 10669

This theorem is referenced by:  ltrel  10692  dfle2  12528  dflt2  12529  itg2gt0cn  35112