Theorem ltrelpr 10889

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.

Step	Hyp	Ref	Expression
1		df-ltp 10876	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
2		opabssxp 5706	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P)
3	1, 2	eqsstri 3976	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 395   ∈ wcel 2111   ⊆ wss 3897   ⊊ wpss 3898  {copab 5151   × cxp 5612  Pcnp 10750  <_P cltp 10754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-ss 3914  df-opab 5152  df-xp 5620  df-ltp 10876

This theorem is referenced by:  ltexpri  10934  ltaprlem  10935  ltapr  10936  suplem1pr  10943  suplem2pr  10944  supexpr  10945  ltsrpr  10968  ltsosr  10985  mappsrpr  10999