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Theorem ltrelpr 11030
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 11017 . 2 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
2 opabssxp 5765 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)} ⊆ (P × P)
31, 2eqsstri 4014 1 <P ⊆ (P × P)
Colors of variables: wff setvar class
Syntax hints:  wa 394  wcel 2099  wss 3947  wpss 3948  {copab 5206   × cxp 5671  Pcnp 10891  <P cltp 10895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-ss 3964  df-opab 5207  df-xp 5679  df-ltp 11017
This theorem is referenced by:  ltexpri  11075  ltaprlem  11076  ltapr  11077  suplem1pr  11084  suplem2pr  11085  supexpr  11086  ltsrpr  11109  ltsosr  11126  mappsrpr  11140
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