Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltrelxr GIF version

Theorem ltrelxr 6877
 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 6862 . 2 < = ({⟨x, y⟩ ∣ (x y x < y)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2 df-3an 886 . . . . . 6 ((x y x < y) ↔ ((x y ℝ) x < y))
32opabbii 3815 . . . . 5 {⟨x, y⟩ ∣ (x y x < y)} = {⟨x, y⟩ ∣ ((x y ℝ) x < y)}
4 opabssxp 4357 . . . . 5 {⟨x, y⟩ ∣ ((x y ℝ) x < y)} ⊆ (ℝ × ℝ)
53, 4eqsstri 2969 . . . 4 {⟨x, y⟩ ∣ (x y x < y)} ⊆ (ℝ × ℝ)
6 rexpssxrxp 6867 . . . 4 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
75, 6sstri 2948 . . 3 {⟨x, y⟩ ∣ (x y x < y)} ⊆ (ℝ* × ℝ*)
8 ressxr 6866 . . . . . 6 ℝ ⊆ ℝ*
9 snsspr2 3504 . . . . . . 7 {-∞} ⊆ {+∞, -∞}
10 ssun2 3101 . . . . . . . 8 {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞})
11 df-xr 6861 . . . . . . . 8 * = (ℝ ∪ {+∞, -∞})
1210, 11sseqtr4i 2972 . . . . . . 7 {+∞, -∞} ⊆ ℝ*
139, 12sstri 2948 . . . . . 6 {-∞} ⊆ ℝ*
148, 13unssi 3112 . . . . 5 (ℝ ∪ {-∞}) ⊆ ℝ*
15 snsspr1 3503 . . . . . 6 {+∞} ⊆ {+∞, -∞}
1615, 12sstri 2948 . . . . 5 {+∞} ⊆ ℝ*
17 xpss12 4388 . . . . 5 (((ℝ ∪ {-∞}) ⊆ ℝ* {+∞} ⊆ ℝ*) → ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*))
1814, 16, 17mp2an 402 . . . 4 ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*)
19 xpss12 4388 . . . . 5 (({-∞} ⊆ ℝ* ℝ ⊆ ℝ*) → ({-∞} × ℝ) ⊆ (ℝ* × ℝ*))
2013, 8, 19mp2an 402 . . . 4 ({-∞} × ℝ) ⊆ (ℝ* × ℝ*)
2118, 20unssi 3112 . . 3 (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ⊆ (ℝ* × ℝ*)
227, 21unssi 3112 . 2 ({⟨x, y⟩ ∣ (x y x < y)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))) ⊆ (ℝ* × ℝ*)
231, 22eqsstri 2969 1 < ⊆ (ℝ* × ℝ*)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ∧ w3a 884   ∈ wcel 1390   ∪ cun 2909   ⊆ wss 2911  {csn 3367  {cpr 3368   class class class wbr 3755  {copab 3808   × cxp 4286  ℝcr 6710   <ℝ cltrr 6715  +∞cpnf 6854  -∞cmnf 6855  ℝ*cxr 6856   < clt 6857 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pr 3374  df-opab 3810  df-xp 4294  df-xr 6861  df-ltxr 6862 This theorem is referenced by:  ltrel  6878
 Copyright terms: Public domain W3C validator