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Theorem ltrelsr 7205
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 7197 . 2 <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
2 opabssxp 4473 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))} ⊆ (R × R)
31, 2eqsstri 3042 1 <R ⊆ (R × R)
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1287  wex 1424  wcel 1436  wss 2986  cop 3428   class class class wbr 3814  {copab 3867   × cxp 4402  (class class class)co 5594  [cec 6223   +P cpp 6773  <P cltp 6775   ~R cer 6776  Rcnr 6777   <R cltr 6783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-in 2992  df-ss 2999  df-opab 3869  df-xp 4410  df-ltr 7197
This theorem is referenced by:  gt0srpr  7215  recexgt0sr  7240  addgt0sr  7242  mulgt0sr  7244  caucvgsrlemcl  7255  caucvgsrlemasr  7256  caucvgsrlemfv  7257  ltresr  7297  axpre-ltirr  7338  axpre-lttrn  7340
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