Theorem enrer

Description: The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of Gleason p. 126. (Contributed by NM, 3-Sep-1995) (Revised by Mario Carneiro, 6-Jul-2015) (New usage is discouraged.)

	Ref	Expression
Assertion	enrer	⊢ ~R Er ( P × P )

Proof

Step	Hyp	Ref	Expression
1		df-enr	⊢ ~R = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ ( P × P ) ∧ 𝑦 ∈ ( P × P ) ) ∧ ∃ 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 ) ) ) }
2		addcompr	⊢ ( 𝑥 +P 𝑦 ) = ( 𝑦 +P 𝑥 )
3		addclpr	⊢ ( ( 𝑥 ∈ P ∧ 𝑦 ∈ P ) → ( 𝑥 +P 𝑦 ) ∈ P )
4		addasspr	⊢ ( ( 𝑥 +P 𝑦 ) +P 𝑧 ) = ( 𝑥 +P ( 𝑦 +P 𝑧 ) )
5		addcanpr	⊢ ( ( 𝑥 ∈ P ∧ 𝑦 ∈ P ) → ( ( 𝑥 +P 𝑦 ) = ( 𝑥 +P 𝑧 ) → 𝑦 = 𝑧 ) )
6	1 2 3 4 5	ecopover	⊢ ~R Er ( P × P )