Theorem ltrelpr 10927

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 10914	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
2		opabssxp 5723	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P)
3	1, 2	eqsstri 3990	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 395   ∈ wcel 2109   ⊆ wss 3911   ⊊ wpss 3912  {copab 5164   × cxp 5629  Pcnp 10788  <_P cltp 10792

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 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-ss 3928  df-opab 5165  df-xp 5637  df-ltp 10914

This theorem is referenced by:  ltexpri  10972  ltaprlem  10973  ltapr  10974  suplem1pr  10981  suplem2pr  10982  supexpr  10983  ltsrpr  11006  ltsosr  11023  mappsrpr  11037