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Theorem ltrelre 11111
Description: 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
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 11105 . 2 < = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
2 opabssxp 5760 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))} ⊆ (ℝ × ℝ)
31, 2eqsstri 4012 1 < ⊆ (ℝ × ℝ)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wex 1781  wcel 2106  wss 3944  cop 4628   class class class wbr 5141  {copab 5203   × cxp 5667  0Rc0r 10843   <R cltr 10848  cr 11091   < cltrr 11096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3951  df-ss 3961  df-opab 5204  df-xp 5675  df-lt 11105
This theorem is referenced by:  ltresr  11117
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