Theorem ltrelre 7001

Description: 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)

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 6994	. 2 ⊢ <ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
2		opabssxp 4432	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))} ⊆ (ℝ × ℝ)
3	1, 2	eqsstri 3029	1 ⊢ <ℝ ⊆ (ℝ × ℝ)

Syntax hints:   ∧ wa 102   = wceq 1284  ∃wex 1421   ∈ wcel 1433   ⊆ wss 2973  ⟨cop 3401   class class class wbr 3785  {copab 3838   × cxp 4361  0_Rc0r 6488   <_R cltr 6493  ℝcr 6980   <_ℝ cltrr 6985

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-in 2979  df-ss 2986  df-opab 3840  df-xp 4369  df-lt 6994

This theorem is referenced by:  ltresr  7007