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Theorem ltrelpr 10109
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 10096 . 2 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
2 opabssxp 5399 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)} ⊆ (P × P)
31, 2eqsstri 3832 1 <P ⊆ (P × P)
Colors of variables: wff setvar class
Syntax hints:  wa 385  wcel 2157  wss 3770  wpss 3771  {copab 4906   × cxp 5311  Pcnp 9970  <P cltp 9974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-in 3777  df-ss 3784  df-opab 4907  df-xp 5319  df-ltp 10096
This theorem is referenced by:  ltexpri  10154  ltaprlem  10155  ltapr  10156  suplem1pr  10163  suplem2pr  10164  supexpr  10165  ltsrpr  10187  ltsosr  10204  mappsrpr  10218
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