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Theorem ltrelpr 9764
Description: Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltrelpr <P ⊆ (P × P)

Proof of Theorem ltrelpr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltp 9751 . 2 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
2 opabssxp 5154 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)} ⊆ (P × P)
31, 2eqsstri 3614 1 <P ⊆ (P × P)
Colors of variables: wff setvar class
Syntax hints:  wa 384  wcel 1987  wss 3555  wpss 3556  {copab 4672   × cxp 5072  Pcnp 9625  <P cltp 9629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-in 3562  df-ss 3569  df-opab 4674  df-xp 5080  df-ltp 9751
This theorem is referenced by:  ltexpri  9809  ltaprlem  9810  ltapr  9811  suplem1pr  9818  suplem2pr  9819  supexpr  9820  ltsrpr  9842  ltsosr  9859  mappsrpr  9873
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