Theorem ltrelpr 11030

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 11017	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
2		opabssxp 5765	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P)
3	1, 2	eqsstri 4014	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 394   ∈ wcel 2099   ⊆ wss 3947   ⊊ wpss 3948  {copab 5206   × cxp 5671  Pcnp 10891  <_P cltp 10895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-ss 3964  df-opab 5207  df-xp 5679  df-ltp 11017

This theorem is referenced by:  ltexpri  11075  ltaprlem  11076  ltapr  11077  suplem1pr  11084  suplem2pr  11085  supexpr  11086  ltsrpr  11109  ltsosr  11126  mappsrpr  11140