Theorem ltrelsr 11166

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 11157	. 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 5748	. 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 3959	1 ⊢ <R ⊆ (R × R)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ⊆ wss 3900  ⟨cop 4838   class class class wbr 5106  {copab 5170   × cxp 5653  (class class class)co 7430  [cec 8729   +_P cpp 10959  <_P cltp 10961   ~_R cer 10962  Rcnr 10963   <_R cltr 10969

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-ss 3901  df-opab 5171  df-xp 5661  df-ltr 11157

This theorem is used by:  ltsrpr  11175  ltasr  11198  recexsrlem  11201  addgt0sr  11202  mulgt0sr  11203  map2psrpr  11208  supsrlem  11209  supsr  11210  ltresr  11238  axpre-lttrn  11264