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Theorem ltrelpr 10273
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 10260 . 2 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
2 opabssxp 5536 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)} ⊆ (P × P)
31, 2eqsstri 3928 1 <P ⊆ (P × P)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wcel 2083  wss 3865  wpss 3866  {copab 5030   × cxp 5448  Pcnp 10134  <P cltp 10138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-in 3872  df-ss 3880  df-opab 5031  df-xp 5456  df-ltp 10260
This theorem is referenced by:  ltexpri  10318  ltaprlem  10319  ltapr  10320  suplem1pr  10327  suplem2pr  10328  supexpr  10329  ltsrpr  10352  ltsosr  10369  mappsrpr  10383
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