Theorem ltrelpr 10956

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 10943	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
2		opabssxp 5739	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P)
3	1, 2	eqsstri 3982	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 399   ∈ wcel 2142   ⊆ wss 3904   ⊊ wpss 3905  {copab 5162   × cxp 5645  Pcnp 10817  <_P cltp 10821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-ss 3921  df-opab 5163  df-xp 5653  df-ltp 10943

This theorem is referenced by:  ltexpri  11001  ltaprlem  11002  ltapr  11003  suplem1pr  11010  suplem2pr  11011  supexpr  11012  ltsrpr  11035  ltsosr  11052  mappsrpr  11066