Theorem ltrelsr 7871

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 7863	. 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 4757	. 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 3229	1 ⊢ <R ⊆ (R × R)

Syntax hints:   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2177   ⊆ wss 3170  ⟨cop 3641   class class class wbr 4051  {copab 4112   × cxp 4681  (class class class)co 5957  [cec 6631   +_P cpp 7426  <_P cltp 7428   ~_R cer 7429  Rcnr 7430   <_R cltr 7436

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-in 3176  df-ss 3183  df-opab 4114  df-xp 4689  df-ltr 7863

This theorem is referenced by:  gt0srpr  7881  recexgt0sr  7906  addgt0sr  7908  mulgt0sr  7911  caucvgsrlemcl  7922  caucvgsrlemasr  7923  caucvgsrlemfv  7924  map2psrprg  7938  suplocsrlemb  7939  suplocsrlempr  7940  suplocsrlem  7941  suplocsr  7942  ltresr  7972  axpre-ltirr  8015  axpre-lttrn  8017