Theorem infm3lem

Description: Lemma for infm3 . (Contributed by NM, 14-Jun-2005)

	Ref	Expression
Assertion	infm3lem	⊢ ( 𝑥 ∈ ℝ → ∃ 𝑦 ∈ ℝ 𝑥 = - 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		renegcl	⊢ ( 𝑥 ∈ ℝ → - 𝑥 ∈ ℝ )
2		recn	⊢ ( 𝑥 ∈ ℝ → 𝑥 ∈ ℂ )
3	2	negnegd	⊢ ( 𝑥 ∈ ℝ → - - 𝑥 = 𝑥 )
4	3	eqcomd	⊢ ( 𝑥 ∈ ℝ → 𝑥 = - - 𝑥 )
5		negeq	⊢ ( 𝑦 = - 𝑥 → - 𝑦 = - - 𝑥 )
6	5	rspceeqv	⊢ ( ( - 𝑥 ∈ ℝ ∧ 𝑥 = - - 𝑥 ) → ∃ 𝑦 ∈ ℝ 𝑥 = - 𝑦 )
7	1 4 6	syl2anc	⊢ ( 𝑥 ∈ ℝ → ∃ 𝑦 ∈ ℝ 𝑥 = - 𝑦 )