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Metamath Proof Explorer

Theorem eqresr

Description: Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	eqresr.1	$⊢ A \in V$
	Assertion	eqresr	$⊢ ⟨A, 0_{𝑹}⟩ = ⟨B, 0_{𝑹}⟩ \leftrightarrow A = B$

Step	Hyp	Ref	Expression
1		eqresr.1	$⊢ A \in V$
2		eqid	$⊢ 0_{𝑹} = 0_{𝑹}$
3		0r	$⊢ 0_{𝑹} \in 𝑹$
4	3	elexi	$⊢ 0_{𝑹} \in V$
5	1 4	opth	$⊢ ⟨A, 0_{𝑹}⟩ = ⟨B, 0_{𝑹}⟩ \leftrightarrow (A = B \land 0_{𝑹} = 0_{𝑹})$
6	2 5	mpbiran2	$⊢ ⟨A, 0_{𝑹}⟩ = ⟨B, 0_{𝑹}⟩ \leftrightarrow A = B$