Theorem ltrelre

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

	Ref	Expression
Assertion	ltrelre	⊢ <ℝ ⊆ ( ℝ × ℝ )

Proof

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