Theorem ltrelre 11232

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 × ℝ)

Proof of Theorem ltrelre

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lt 11226	. 2 ⊢ <ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
2		opabssxp 5748	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))} ⊆ (ℝ × ℝ)
3	1, 2	eqsstri 3959	1 ⊢ <ℝ ⊆ (ℝ × ℝ)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ⊆ wss 3900  ⟨cop 4838   class class class wbr 5106  {copab 5170   × cxp 5653  0_Rc0r 10964   <_R cltr 10969  ℝcr 11212   <_ℝ cltrr 11217

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-ss 3901  df-opab 5171  df-xp 5661  df-lt 11226

This theorem is used by:  ltresr  11238