Theorem ltrelsr 11087

Description: Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
ltrelsr	⊢ <R ⊆ (R × R)

Proof of Theorem ltrelsr

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ltr 11078	. 2 ⊢ <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
2		opabssxp 5752	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))} ⊆ (R × R)
3	1, 2	eqsstri 4010	1 ⊢ <R ⊆ (R × R)

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ⊆ wss 3931  ⟨cop 4612   class class class wbr 5124  {copab 5186   × cxp 5657  (class class class)co 7410  [cec 8722   +_P cpp 10880  <_P cltp 10882   ~_R cer 10883  Rcnr 10884   <_R cltr 10890

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-ltr 11078

This theorem is referenced by:  ltsrpr  11096  ltasr  11119  recexsrlem  11122  addgt0sr  11123  mulgt0sr  11124  map2psrpr  11129  supsrlem  11130  supsr  11131  ltresr  11159  axpre-lttrn  11185