Theorem ltrelsr

Description: Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		df-ltr	⊢ <R = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ R ∧ 𝑦 ∈ R ) ∧ ∃ 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R ∧ 𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R ) ∧ ( 𝑧 +P 𝑢 ) <P ( 𝑤 +P 𝑣 ) ) ) }
2		opabssxp	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ R ∧ 𝑦 ∈ R ) ∧ ∃ 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R ∧ 𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R ) ∧ ( 𝑧 +P 𝑢 ) <P ( 𝑤 +P 𝑣 ) ) ) } ⊆ ( R × R )
3	1 2	eqsstri	⊢ <R ⊆ ( R × R )