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Theorem ltrelsr 7693
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 7685 . 2 <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
2 opabssxp 4683 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))} ⊆ (R × R)
31, 2eqsstri 3179 1 <R ⊆ (R × R)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1348  wex 1485  wcel 2141  wss 3121  cop 3584   class class class wbr 3987  {copab 4047   × cxp 4607  (class class class)co 5851  [cec 6509   +P cpp 7248  <P cltp 7250   ~R cer 7251  Rcnr 7252   <R cltr 7258
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-opab 4049  df-xp 4615  df-ltr 7685
This theorem is referenced by:  gt0srpr  7703  recexgt0sr  7728  addgt0sr  7730  mulgt0sr  7733  caucvgsrlemcl  7744  caucvgsrlemasr  7745  caucvgsrlemfv  7746  map2psrprg  7760  suplocsrlemb  7761  suplocsrlempr  7762  suplocsrlem  7763  suplocsr  7764  ltresr  7794  axpre-ltirr  7837  axpre-lttrn  7839
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