Theorem ltrelpr 11039

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 11026	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
2		opabssxp 5777	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P)
3	1, 2	eqsstri 4029	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 395   ∈ wcel 2107   ⊆ wss 3950   ⊊ wpss 3951  {copab 5204   × cxp 5682  Pcnp 10900  <_P cltp 10904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-ss 3967  df-opab 5205  df-xp 5690  df-ltp 11026

This theorem is referenced by:  ltexpri  11084  ltaprlem  11085  ltapr  11086  suplem1pr  11093  suplem2pr  11094  supexpr  11095  ltsrpr  11118  ltsosr  11135  mappsrpr  11149