Theorem ltrelsr 7805

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 7797	. 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 4737	. 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 3215	1 ⊢ <R ⊆ (R × R)

Syntax hints:   ∧ wa 104   = wceq 1364  ∃wex 1506   ∈ wcel 2167   ⊆ wss 3157  ⟨cop 3625   class class class wbr 4033  {copab 4093   × cxp 4661  (class class class)co 5922  [cec 6590   +_P cpp 7360  <_P cltp 7362   ~_R cer 7363  Rcnr 7364   <_R cltr 7370

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-in 3163  df-ss 3170  df-opab 4095  df-xp 4669  df-ltr 7797

This theorem is referenced by:  gt0srpr  7815  recexgt0sr  7840  addgt0sr  7842  mulgt0sr  7845  caucvgsrlemcl  7856  caucvgsrlemasr  7857  caucvgsrlemfv  7858  map2psrprg  7872  suplocsrlemb  7873  suplocsrlempr  7874  suplocsrlem  7875  suplocsr  7876  ltresr  7906  axpre-ltirr  7949  axpre-lttrn  7951