Theorem ltrelsr 11106

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 11097	. 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 5781	. 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 4030	1 ⊢ <R ⊆ (R × R)

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ⊆ wss 3963  ⟨cop 4637   class class class wbr 5148  {copab 5210   × cxp 5687  (class class class)co 7431  [cec 8742   +_P cpp 10899  <_P cltp 10901   ~_R cer 10902  Rcnr 10903   <_R cltr 10909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-ss 3980  df-opab 5211  df-xp 5695  df-ltr 11097

This theorem is referenced by:  ltsrpr  11115  ltasr  11138  recexsrlem  11141  addgt0sr  11142  mulgt0sr  11143  map2psrpr  11148  supsrlem  11149  supsr  11150  ltresr  11178  axpre-lttrn  11204