Theorem ltrelxr 11279

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 11257	. 2 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2		df-3an 1087	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦))
3	2	opabbii 5214	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)}
4		opabssxp 5767	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ × ℝ)
5	3, 4	eqsstri 4015	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ × ℝ)
6		rexpssxrxp 11263	. . . 4 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
7	5, 6	sstri 3990	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ* × ℝ*)
8		ressxr 11262	. . . . . 6 ⊢ ℝ ⊆ ℝ*
9		snsspr2 4817	. . . . . . 7 ⊢ {-∞} ⊆ {+∞, -∞}
10		ssun2 4172	. . . . . . . 8 ⊢ {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞})
11		df-xr 11256	. . . . . . . 8 ⊢ ℝ* = (ℝ ∪ {+∞, -∞})
12	10, 11	sseqtrri 4018	. . . . . . 7 ⊢ {+∞, -∞} ⊆ ℝ*
13	9, 12	sstri 3990	. . . . . 6 ⊢ {-∞} ⊆ ℝ*
14	8, 13	unssi 4184	. . . . 5 ⊢ (ℝ ∪ {-∞}) ⊆ ℝ*
15		snsspr1 4816	. . . . . 6 ⊢ {+∞} ⊆ {+∞, -∞}
16	15, 12	sstri 3990	. . . . 5 ⊢ {+∞} ⊆ ℝ*
17		xpss12 5690	. . . . 5 ⊢ (((ℝ ∪ {-∞}) ⊆ ℝ* ∧ {+∞} ⊆ ℝ*) → ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*))
18	14, 16, 17	mp2an 688	. . . 4 ⊢ ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*)
19		xpss12 5690	. . . . 5 ⊢ (({-∞} ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ({-∞} × ℝ) ⊆ (ℝ* × ℝ*))
20	13, 8, 19	mp2an 688	. . . 4 ⊢ ({-∞} × ℝ) ⊆ (ℝ* × ℝ*)
21	18, 20	unssi 4184	. . 3 ⊢ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ⊆ (ℝ* × ℝ*)
22	7, 21	unssi 4184	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))) ⊆ (ℝ* × ℝ*)
23	1, 22	eqsstri 4015	1 ⊢ < ⊆ (ℝ* × ℝ*)

Syntax hints:   ∧ wa 394   ∧ w3a 1085   ∈ wcel 2104   ∪ cun 3945   ⊆ wss 3947  {csn 4627  {cpr 4629   class class class wbr 5147  {copab 5209   × cxp 5673  ℝcr 11111   <_ℝ cltrr 11116  +∞cpnf 11249  -∞cmnf 11250  ℝ^*cxr 11251   < clt 11252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-un 3952  df-in 3954  df-ss 3964  df-pr 4630  df-opab 5210  df-xp 5681  df-xr 11256  df-ltxr 11257

This theorem is referenced by:  ltrel  11280  dfle2  13130  dflt2  13131  itg2gt0cn  36846  et-ltneverrefl  45885  natglobalincr  45889  iccdisj2  47617