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Theorem ltrelsr 10081
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 10073 . 2 <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
2 opabssxp 5350 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))} ⊆ (R × R)
31, 2eqsstri 3776 1 <R ⊆ (R × R)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  wss 3715  cop 4327   class class class wbr 4804  {copab 4864   × cxp 5264  (class class class)co 6813  [cec 7909   +P cpp 9875  <P cltp 9877   ~R cer 9878  Rcnr 9879   <R cltr 9885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-in 3722  df-ss 3729  df-opab 4865  df-xp 5272  df-ltr 10073
This theorem is referenced by:  ltsrpr  10090  ltasr  10113  recexsrlem  10116  addgt0sr  10117  mulgt0sr  10118  map2psrpr  10123  supsrlem  10124  supsr  10125  ltresr  10153  axpre-lttrn  10179
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