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Theorem ltrelre 11174
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 11168 . 2 < = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
2 opabssxp 5778 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))} ⊆ (ℝ × ℝ)
31, 2eqsstri 4030 1 < ⊆ (ℝ × ℝ)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wex 1779  wcel 2108  wss 3951  cop 4632   class class class wbr 5143  {copab 5205   × cxp 5683  0Rc0r 10906   <R cltr 10911  cr 11154   < cltrr 11159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-ss 3968  df-opab 5206  df-xp 5691  df-lt 11168
This theorem is referenced by:  ltresr  11180
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