Theorem ltrelpr 11017

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 11004	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
2		opabssxp 5752	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P)
3	1, 2	eqsstri 4010	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 395   ∈ wcel 2109   ⊆ wss 3931   ⊊ wpss 3932  {copab 5186   × cxp 5657  Pcnp 10878  <_P cltp 10882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-ss 3948  df-opab 5187  df-xp 5665  df-ltp 11004

This theorem is referenced by:  ltexpri  11062  ltaprlem  11063  ltapr  11064  suplem1pr  11071  suplem2pr  11072  supexpr  11073  ltsrpr  11096  ltsosr  11113  mappsrpr  11127