Theorem ltrelsr 8105

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 8097	. 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 4849	. 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 3280	1 ⊢ <R ⊆ (R × R)

This proof depends on syntax axioms:   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ⊆ wss 3220  ⟨cop 3712   class class class wbr 4130  {copab 4191   × cxp 4772  (class class class)co 6085  [cec 6805   +_P cpp 7660  <_P cltp 7662   ~_R cer 7663  Rcnr 7664   <_R cltr 7670

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233  df-opab 4193  df-xp 4780  df-ltr 8097

This theorem is used by:  gt0srpr  8115  recexgt0sr  8140  addgt0sr  8142  mulgt0sr  8145  caucvgsrlemcl  8156  caucvgsrlemasr  8157  caucvgsrlemfv  8158  map2psrprg  8172  suplocsrlemb  8173  suplocsrlempr  8174  suplocsrlem  8175  suplocsr  8176  ltresr  8206  axpre-ltirr  8249  axpre-lttrn  8251