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Theorem ltrelxr 10355
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 10335 . 2 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2 df-3an 1109 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦))
32opabbii 4878 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦)}
4 opabssxp 5365 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦)} ⊆ (ℝ × ℝ)
53, 4eqsstri 3797 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ⊆ (ℝ × ℝ)
6 rexpssxrxp 10340 . . . 4 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
75, 6sstri 3772 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ⊆ (ℝ* × ℝ*)
8 ressxr 10339 . . . . . 6 ℝ ⊆ ℝ*
9 snsspr2 4502 . . . . . . 7 {-∞} ⊆ {+∞, -∞}
10 ssun2 3941 . . . . . . . 8 {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞})
11 df-xr 10334 . . . . . . . 8 * = (ℝ ∪ {+∞, -∞})
1210, 11sseqtr4i 3800 . . . . . . 7 {+∞, -∞} ⊆ ℝ*
139, 12sstri 3772 . . . . . 6 {-∞} ⊆ ℝ*
148, 13unssi 3952 . . . . 5 (ℝ ∪ {-∞}) ⊆ ℝ*
15 snsspr1 4501 . . . . . 6 {+∞} ⊆ {+∞, -∞}
1615, 12sstri 3772 . . . . 5 {+∞} ⊆ ℝ*
17 xpss12 5294 . . . . 5 (((ℝ ∪ {-∞}) ⊆ ℝ* ∧ {+∞} ⊆ ℝ*) → ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*))
1814, 16, 17mp2an 683 . . . 4 ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*)
19 xpss12 5294 . . . . 5 (({-∞} ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ({-∞} × ℝ) ⊆ (ℝ* × ℝ*))
2013, 8, 19mp2an 683 . . . 4 ({-∞} × ℝ) ⊆ (ℝ* × ℝ*)
2118, 20unssi 3952 . . 3 (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ⊆ (ℝ* × ℝ*)
227, 21unssi 3952 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))) ⊆ (ℝ* × ℝ*)
231, 22eqsstri 3797 1 < ⊆ (ℝ* × ℝ*)
Colors of variables: wff setvar class
Syntax hints:  wa 384  w3a 1107  wcel 2155  cun 3732  wss 3734  {csn 4336  {cpr 4338   class class class wbr 4811  {copab 4873   × cxp 5277  cr 10190   < cltrr 10195  +∞cpnf 10327  -∞cmnf 10328  *cxr 10329   < clt 10330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3739  df-in 3741  df-ss 3748  df-pr 4339  df-opab 4874  df-xp 5285  df-xr 10334  df-ltxr 10335
This theorem is referenced by:  ltrel  10356  dfle2  12183  dflt2  12184  itg2gt0cn  33891
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