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Theorem ltrelxr 11262
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 11240 . 2 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2 df-3an 1090 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦))
32opabbii 5211 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦)}
4 opabssxp 5763 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦)} ⊆ (ℝ × ℝ)
53, 4eqsstri 4014 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ⊆ (ℝ × ℝ)
6 rexpssxrxp 11246 . . . 4 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
75, 6sstri 3989 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ⊆ (ℝ* × ℝ*)
8 ressxr 11245 . . . . . 6 ℝ ⊆ ℝ*
9 snsspr2 4814 . . . . . . 7 {-∞} ⊆ {+∞, -∞}
10 ssun2 4171 . . . . . . . 8 {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞})
11 df-xr 11239 . . . . . . . 8 * = (ℝ ∪ {+∞, -∞})
1210, 11sseqtrri 4017 . . . . . . 7 {+∞, -∞} ⊆ ℝ*
139, 12sstri 3989 . . . . . 6 {-∞} ⊆ ℝ*
148, 13unssi 4183 . . . . 5 (ℝ ∪ {-∞}) ⊆ ℝ*
15 snsspr1 4813 . . . . . 6 {+∞} ⊆ {+∞, -∞}
1615, 12sstri 3989 . . . . 5 {+∞} ⊆ ℝ*
17 xpss12 5687 . . . . 5 (((ℝ ∪ {-∞}) ⊆ ℝ* ∧ {+∞} ⊆ ℝ*) → ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*))
1814, 16, 17mp2an 691 . . . 4 ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*)
19 xpss12 5687 . . . . 5 (({-∞} ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ({-∞} × ℝ) ⊆ (ℝ* × ℝ*))
2013, 8, 19mp2an 691 . . . 4 ({-∞} × ℝ) ⊆ (ℝ* × ℝ*)
2118, 20unssi 4183 . . 3 (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ⊆ (ℝ* × ℝ*)
227, 21unssi 4183 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))) ⊆ (ℝ* × ℝ*)
231, 22eqsstri 4014 1 < ⊆ (ℝ* × ℝ*)
Colors of variables: wff setvar class
Syntax hints:  wa 397  w3a 1088  wcel 2107  cun 3944  wss 3946  {csn 4624  {cpr 4626   class class class wbr 5144  {copab 5206   × cxp 5670  cr 11096   < cltrr 11101  +∞cpnf 11232  -∞cmnf 11233  *cxr 11234   < clt 11235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3951  df-in 3953  df-ss 3963  df-pr 4627  df-opab 5207  df-xp 5678  df-xr 11239  df-ltxr 11240
This theorem is referenced by:  ltrel  11263  dfle2  13113  dflt2  13114  itg2gt0cn  36448  et-ltneverrefl  45460  natglobalincr  45464  iccdisj2  47370
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