Theorem ltrelpr 10824

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 10811	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
2		opabssxp 5695	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P)
3	1, 2	eqsstri 3964	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 396   ∈ wcel 2105   ⊆ wss 3896   ⊊ wpss 3897  {copab 5147   × cxp 5603  Pcnp 10685  <_P cltp 10689

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3443  df-in 3903  df-ss 3913  df-opab 5148  df-xp 5611  df-ltp 10811

This theorem is referenced by:  ltexpri  10869  ltaprlem  10870  ltapr  10871  suplem1pr  10878  suplem2pr  10879  supexpr  10880  ltsrpr  10903  ltsosr  10920  mappsrpr  10934