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Theorem ltrelpr 10408
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 10395 . 2 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
2 opabssxp 5636 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)} ⊆ (P × P)
31, 2eqsstri 3998 1 <P ⊆ (P × P)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wcel 2105  wss 3933  wpss 3934  {copab 5119   × cxp 5546  Pcnp 10269  <P cltp 10273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-in 3940  df-ss 3949  df-opab 5120  df-xp 5554  df-ltp 10395
This theorem is referenced by:  ltexpri  10453  ltaprlem  10454  ltapr  10455  suplem1pr  10462  suplem2pr  10463  supexpr  10464  ltsrpr  10487  ltsosr  10504  mappsrpr  10518
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