Theorem ltrelsr 11153

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 11144	. 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 5743	. 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 3977	1 ⊢ <R ⊆ (R × R)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ⊆ wss 3899  ⟨cop 4590   class class class wbr 5103  {copab 5167   × cxp 5649  (class class class)co 7420  [cec 8715   +_P cpp 10946  <_P cltp 10948   ~_R cer 10949  Rcnr 10950   <_R cltr 10956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-ss 3916  df-opab 5168  df-xp 5657  df-ltr 11144

This theorem is used by:  ltsrpr  11162  ltasr  11185  recexsrlem  11188  addgt0sr  11189  mulgt0sr  11190  map2psrpr  11195  supsrlem  11196  supsr  11197  ltresr  11225  axpre-lttrn  11251