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Theorem ltrelpr 9805
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 9792 . 2 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
2 opabssxp 5183 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)} ⊆ (P × P)
31, 2eqsstri 3627 1 <P ⊆ (P × P)
Colors of variables: wff setvar class
Syntax hints:  wa 384  wcel 1988  wss 3567  wpss 3568  {copab 4703   × cxp 5102  Pcnp 9666  <P cltp 9670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-in 3574  df-ss 3581  df-opab 4704  df-xp 5110  df-ltp 9792
This theorem is referenced by:  ltexpri  9850  ltaprlem  9851  ltapr  9852  suplem1pr  9859  suplem2pr  9860  supexpr  9861  ltsrpr  9883  ltsosr  9900  mappsrpr  9914
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