Theorem ltrelsr 11027

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 11018	. 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 5740	. 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 3983	1 ⊢ <R ⊆ (R × R)

Syntax hints:   ∧ wa 399   = wceq 1561  ∃wex 1800   ∈ wcel 2143   ⊆ wss 3905  ⟨cop 4589   class class class wbr 5101  {copab 5163   × cxp 5646  (class class class)co 7397  [cec 8677   +_P cpp 10820  <_P cltp 10822   ~_R cer 10823  Rcnr 10824   <_R cltr 10830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-ss 3922  df-opab 5164  df-xp 5654  df-ltr 11018

This theorem is referenced by:  ltsrpr  11036  ltasr  11059  recexsrlem  11062  addgt0sr  11063  mulgt0sr  11064  map2psrpr  11069  supsrlem  11070  supsr  11071  ltresr  11099  axpre-lttrn  11125