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Theorem ltrelpr 10800
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 10787 . 2 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
2 opabssxp 5690 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)} ⊆ (P × P)
31, 2eqsstri 3960 1 <P ⊆ (P × P)
Colors of variables: wff setvar class
Syntax hints:  wa 397  wcel 2104  wss 3892  wpss 3893  {copab 5143   × cxp 5598  Pcnp 10661  <P cltp 10665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-in 3899  df-ss 3909  df-opab 5144  df-xp 5606  df-ltp 10787
This theorem is referenced by:  ltexpri  10845  ltaprlem  10846  ltapr  10847  suplem1pr  10854  suplem2pr  10855  supexpr  10856  ltsrpr  10879  ltsosr  10896  mappsrpr  10910
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