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Theorem ltrelxr 10753
 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 10731 . 2 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2 df-3an 1086 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦))
32opabbii 5103 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦)}
4 opabssxp 5617 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦)} ⊆ (ℝ × ℝ)
53, 4eqsstri 3928 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ⊆ (ℝ × ℝ)
6 rexpssxrxp 10737 . . . 4 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
75, 6sstri 3903 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ⊆ (ℝ* × ℝ*)
8 ressxr 10736 . . . . . 6 ℝ ⊆ ℝ*
9 snsspr2 4708 . . . . . . 7 {-∞} ⊆ {+∞, -∞}
10 ssun2 4080 . . . . . . . 8 {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞})
11 df-xr 10730 . . . . . . . 8 * = (ℝ ∪ {+∞, -∞})
1210, 11sseqtrri 3931 . . . . . . 7 {+∞, -∞} ⊆ ℝ*
139, 12sstri 3903 . . . . . 6 {-∞} ⊆ ℝ*
148, 13unssi 4092 . . . . 5 (ℝ ∪ {-∞}) ⊆ ℝ*
15 snsspr1 4707 . . . . . 6 {+∞} ⊆ {+∞, -∞}
1615, 12sstri 3903 . . . . 5 {+∞} ⊆ ℝ*
17 xpss12 5543 . . . . 5 (((ℝ ∪ {-∞}) ⊆ ℝ* ∧ {+∞} ⊆ ℝ*) → ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*))
1814, 16, 17mp2an 691 . . . 4 ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*)
19 xpss12 5543 . . . . 5 (({-∞} ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ({-∞} × ℝ) ⊆ (ℝ* × ℝ*))
2013, 8, 19mp2an 691 . . . 4 ({-∞} × ℝ) ⊆ (ℝ* × ℝ*)
2118, 20unssi 4092 . . 3 (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ⊆ (ℝ* × ℝ*)
227, 21unssi 4092 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))) ⊆ (ℝ* × ℝ*)
231, 22eqsstri 3928 1 < ⊆ (ℝ* × ℝ*)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111   ∪ cun 3858   ⊆ wss 3860  {csn 4525  {cpr 4527   class class class wbr 5036  {copab 5098   × cxp 5526  ℝcr 10587   <ℝ cltrr 10592  +∞cpnf 10723  -∞cmnf 10724  ℝ*cxr 10725   < clt 10726 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-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-pr 4528  df-opab 5099  df-xp 5534  df-xr 10730  df-ltxr 10731 This theorem is referenced by:  ltrel  10754  dfle2  12594  dflt2  12595  itg2gt0cn  35426  iccdisj2  45632
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