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

Theorem ltrelsr 9927
Description: Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltrelsr <R ⊆ (R × R)

Proof of Theorem ltrelsr
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltr 9919 . 2 <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
2 opabssxp 5227 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))} ⊆ (R × R)
31, 2eqsstri 3668 1 <R ⊆ (R × R)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wex 1744  wcel 2030  wss 3607  cop 4216   class class class wbr 4685  {copab 4745   × cxp 5141  (class class class)co 6690  [cec 7785   +P cpp 9721  <P cltp 9723   ~R cer 9724  Rcnr 9725   <R cltr 9731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-in 3614  df-ss 3621  df-opab 4746  df-xp 5149  df-ltr 9919
This theorem is referenced by:  ltsrpr  9936  ltasr  9959  recexsrlem  9962  addgt0sr  9963  mulgt0sr  9964  map2psrpr  9969  supsrlem  9970  supsr  9971  ltresr  9999  axpre-lttrn  10025
  Copyright terms: Public domain W3C validator