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Theorem ltrelpr 9699
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 9686 . 2 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
2 opabssxp 5116 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)} ⊆ (P × P)
31, 2eqsstri 3598 1 <P ⊆ (P × P)
Colors of variables: wff setvar class
Syntax hints:  wa 383  wcel 1977  wss 3540  wpss 3541  {copab 4642   × cxp 5036  Pcnp 9560  <P cltp 9564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-in 3547  df-ss 3554  df-opab 4644  df-xp 5044  df-ltp 9686
This theorem is referenced by:  ltexpri  9744  ltaprlem  9745  ltapr  9746  suplem1pr  9753  suplem2pr  9754  supexpr  9755  ltsrpr  9777  ltsosr  9794  mappsrpr  9808
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