MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltrelxr Structured version   Visualization version   GIF version

Theorem ltrelxr 11322
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.
StepHypRef Expression
1 df-ltxr 11300 . 2 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2 df-3an 1089 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦))
32opabbii 5210 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦)}
4 opabssxp 5778 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦)} ⊆ (ℝ × ℝ)
53, 4eqsstri 4030 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ⊆ (ℝ × ℝ)
6 rexpssxrxp 11306 . . . 4 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
75, 6sstri 3993 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ⊆ (ℝ* × ℝ*)
8 ressxr 11305 . . . . . 6 ℝ ⊆ ℝ*
9 snsspr2 4815 . . . . . . 7 {-∞} ⊆ {+∞, -∞}
10 ssun2 4179 . . . . . . . 8 {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞})
11 df-xr 11299 . . . . . . . 8 * = (ℝ ∪ {+∞, -∞})
1210, 11sseqtrri 4033 . . . . . . 7 {+∞, -∞} ⊆ ℝ*
139, 12sstri 3993 . . . . . 6 {-∞} ⊆ ℝ*
148, 13unssi 4191 . . . . 5 (ℝ ∪ {-∞}) ⊆ ℝ*
15 snsspr1 4814 . . . . . 6 {+∞} ⊆ {+∞, -∞}
1615, 12sstri 3993 . . . . 5 {+∞} ⊆ ℝ*
17 xpss12 5700 . . . . 5 (((ℝ ∪ {-∞}) ⊆ ℝ* ∧ {+∞} ⊆ ℝ*) → ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*))
1814, 16, 17mp2an 692 . . . 4 ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*)
19 xpss12 5700 . . . . 5 (({-∞} ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ({-∞} × ℝ) ⊆ (ℝ* × ℝ*))
2013, 8, 19mp2an 692 . . . 4 ({-∞} × ℝ) ⊆ (ℝ* × ℝ*)
2118, 20unssi 4191 . . 3 (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ⊆ (ℝ* × ℝ*)
227, 21unssi 4191 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))) ⊆ (ℝ* × ℝ*)
231, 22eqsstri 4030 1 < ⊆ (ℝ* × ℝ*)
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1087  wcel 2108  cun 3949  wss 3951  {csn 4626  {cpr 4628   class class class wbr 5143  {copab 5205   × cxp 5683  cr 11154   < cltrr 11159  +∞cpnf 11292  -∞cmnf 11293  *cxr 11294   < clt 11295
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-pr 4629  df-opab 5206  df-xp 5691  df-xr 11299  df-ltxr 11300
This theorem is referenced by:  ltrel  11323  dfle2  13189  dflt2  13190  itg2gt0cn  37682  ltex  42286  et-ltneverrefl  46886  natglobalincr  46892  iccdisj2  48795
  Copyright terms: Public domain W3C validator