Theorem ltrelpr 10913

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 10900	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
2		opabssxp 5717	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P)
3	1, 2	eqsstri 3981	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 395   ∈ wcel 2114   ⊆ wss 3902   ⊊ wpss 3903  {copab 5161   × cxp 5623  Pcnp 10774  <_P cltp 10778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-ss 3919  df-opab 5162  df-xp 5631  df-ltp 10900

This theorem is referenced by:  ltexpri  10958  ltaprlem  10959  ltapr  10960  suplem1pr  10967  suplem2pr  10968  supexpr  10969  ltsrpr  10992  ltsosr  11009  mappsrpr  11023