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

Theorem ltrelsr 7558
Description: Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)
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 7550 . 2 <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
2 opabssxp 4613 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))} ⊆ (R × R)
31, 2eqsstri 3129 1 <R ⊆ (R × R)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1331  wex 1468  wcel 1480  wss 3071  cop 3530   class class class wbr 3929  {copab 3988   × cxp 4537  (class class class)co 5774  [cec 6427   +P cpp 7113  <P cltp 7115   ~R cer 7116  Rcnr 7117   <R cltr 7123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-in 3077  df-ss 3084  df-opab 3990  df-xp 4545  df-ltr 7550
This theorem is referenced by:  gt0srpr  7568  recexgt0sr  7593  addgt0sr  7595  mulgt0sr  7598  caucvgsrlemcl  7609  caucvgsrlemasr  7610  caucvgsrlemfv  7611  map2psrprg  7625  suplocsrlemb  7626  suplocsrlempr  7627  suplocsrlem  7628  suplocsr  7629  ltresr  7659  axpre-ltirr  7702  axpre-lttrn  7704
  Copyright terms: Public domain W3C validator