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Theorem ltrelsr 6666
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 6658 . 2 <R = {⟨x, y⟩ ∣ ((x R y R) zwvu((x = [⟨z, w⟩] ~R y = [⟨v, u⟩] ~R ) (z +P u)<P (w +P v)))}
2 opabssxp 4357 . 2 {⟨x, y⟩ ∣ ((x R y R) zwvu((x = [⟨z, w⟩] ~R y = [⟨v, u⟩] ~R ) (z +P u)<P (w +P v)))} ⊆ (R × R)
31, 2eqsstri 2969 1 <R ⊆ (R × R)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  wss 2911  cop 3370   class class class wbr 3755  {copab 3808   × cxp 4286  (class class class)co 5455  [cec 6040   +P cpp 6277  <P cltp 6279   ~R cer 6280  Rcnr 6281   <R cltr 6287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-opab 3810  df-xp 4294  df-ltr 6658
This theorem is referenced by:  gt0srpr  6676  recexgt0sr  6701  addgt0sr  6703  mulgt0sr  6704  ltresr  6736  axpre-ltirr  6766  axpre-lttrn  6768
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