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Theorem ltrelsr 11108
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 11099 . 2 <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
2 opabssxp 5778 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))} ⊆ (R × R)
31, 2eqsstri 4030 1 <R ⊆ (R × R)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wex 1779  wcel 2108  wss 3951  cop 4632   class class class wbr 5143  {copab 5205   × cxp 5683  (class class class)co 7431  [cec 8743   +P cpp 10901  <P cltp 10903   ~R cer 10904  Rcnr 10905   <R cltr 10911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-ss 3968  df-opab 5206  df-xp 5691  df-ltr 11099
This theorem is referenced by:  ltsrpr  11117  ltasr  11140  recexsrlem  11143  addgt0sr  11144  mulgt0sr  11145  map2psrpr  11150  supsrlem  11151  supsr  11152  ltresr  11180  axpre-lttrn  11206
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