Theorem ltrelsr 10959

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 10950	. 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 5706	. 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 3976	1 ⊢ <R ⊆ (R × R)

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111   ⊆ wss 3897  ⟨cop 4579   class class class wbr 5089  {copab 5151   × cxp 5612  (class class class)co 7346  [cec 8620   +_P cpp 10752  <_P cltp 10754   ~_R cer 10755  Rcnr 10756   <_R cltr 10762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-ss 3914  df-opab 5152  df-xp 5620  df-ltr 10950

This theorem is referenced by:  ltsrpr  10968  ltasr  10991  recexsrlem  10994  addgt0sr  10995  mulgt0sr  10996  map2psrpr  11001  supsrlem  11002  supsr  11003  ltresr  11031  axpre-lttrn  11057