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Theorem ltrelre 7810
Description: 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
Assertion
Ref Expression
ltrelre < ⊆ (ℝ × ℝ)

Proof of Theorem ltrelre
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lt 7802 . 2 < = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
2 opabssxp 4696 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))} ⊆ (ℝ × ℝ)
31, 2eqsstri 3187 1 < ⊆ (ℝ × ℝ)
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wex 1492  wcel 2148  wss 3129  cop 3594   class class class wbr 4000  {copab 4060   × cxp 4620  0Rc0r 7275   <R cltr 7280  cr 7788   < cltrr 7793
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3135  df-ss 3142  df-opab 4062  df-xp 4628  df-lt 7802
This theorem is referenced by:  ltresr  7816
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