Theorem 0cnALT2

Description: Alternate proof of 0cnALT which is shorter, but depends on ax-8 , ax-13 , ax-sep , ax-nul , ax-pow , ax-pr , ax-un , and every complex number axiom except ax-pre-mulgt0 and ax-pre-sup . (Contributed by NM, 19-Feb-2005) (Revised by Mario Carneiro, 27-May-2016) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	0cnALT2	⊢ 0 ∈ ℂ

Proof

Step	Hyp	Ref	Expression
1		ax-icn	⊢ i ∈ ℂ
2		cnegex	⊢ ( i ∈ ℂ → ∃ 𝑥 ∈ ℂ ( i + 𝑥 ) = 0 )
3	1 2	ax-mp	⊢ ∃ 𝑥 ∈ ℂ ( i + 𝑥 ) = 0
4		addcl	⊢ ( ( i ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( i + 𝑥 ) ∈ ℂ )
5	1 4	mpan	⊢ ( 𝑥 ∈ ℂ → ( i + 𝑥 ) ∈ ℂ )
6		eleq1	⊢ ( ( i + 𝑥 ) = 0 → ( ( i + 𝑥 ) ∈ ℂ ↔ 0 ∈ ℂ ) )
7	5 6	syl5ibcom	⊢ ( 𝑥 ∈ ℂ → ( ( i + 𝑥 ) = 0 → 0 ∈ ℂ ) )
8	7	rexlimiv	⊢ ( ∃ 𝑥 ∈ ℂ ( i + 𝑥 ) = 0 → 0 ∈ ℂ )
9	3 8	ax-mp	⊢ 0 ∈ ℂ