Theorem ltrelsr 7957

Description: Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)

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 7949	. 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 4800	. 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 3259	1 ⊢ <R ⊆ (R × R)

Syntax hints:   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2202   ⊆ wss 3200  ⟨cop 3672   class class class wbr 4088  {copab 4149   × cxp 4723  (class class class)co 6017  [cec 6699   +_P cpp 7512  <_P cltp 7514   ~_R cer 7515  Rcnr 7516   <_R cltr 7522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-in 3206  df-ss 3213  df-opab 4151  df-xp 4731  df-ltr 7949

This theorem is referenced by:  gt0srpr  7967  recexgt0sr  7992  addgt0sr  7994  mulgt0sr  7997  caucvgsrlemcl  8008  caucvgsrlemasr  8009  caucvgsrlemfv  8010  map2psrprg  8024  suplocsrlemb  8025  suplocsrlempr  8026  suplocsrlem  8027  suplocsr  8028  ltresr  8058  axpre-ltirr  8101  axpre-lttrn  8103