Theorem ltrelsr 7921

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 7913	. 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 4792	. 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 3256	1 ⊢ <R ⊆ (R × R)

Syntax hints:   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200   ⊆ wss 3197  ⟨cop 3669   class class class wbr 4082  {copab 4143   × cxp 4716  (class class class)co 6000  [cec 6676   +_P cpp 7476  <_P cltp 7478   ~_R cer 7479  Rcnr 7480   <_R cltr 7486

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3203  df-ss 3210  df-opab 4145  df-xp 4724  df-ltr 7913

This theorem is referenced by:  gt0srpr  7931  recexgt0sr  7956  addgt0sr  7958  mulgt0sr  7961  caucvgsrlemcl  7972  caucvgsrlemasr  7973  caucvgsrlemfv  7974  map2psrprg  7988  suplocsrlemb  7989  suplocsrlempr  7990  suplocsrlem  7991  suplocsr  7992  ltresr  8022  axpre-ltirr  8065  axpre-lttrn  8067