Theorem ltrelpr 10912

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 10899	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
2		opabssxp 5710	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P)
3	1, 2	eqsstri 3961	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 396   ∈ wcel 2119   ⊆ wss 3883   ⊊ wpss 3884  {copab 5134   × cxp 5616  Pcnp 10773  <_P cltp 10777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-ss 3900  df-opab 5135  df-xp 5624  df-ltp 10899

This theorem is referenced by:  ltexpri  10957  ltaprlem  10958  ltapr  10959  suplem1pr  10966  suplem2pr  10967  supexpr  10968  ltsrpr  10991  ltsosr  11008  mappsrpr  11022