Theorem peano2cn 7148

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4318. (Contributed by NM, 17-Aug-2005.)

Assertion

Ref	Expression
peano2cn	⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Proof of Theorem peano2cn

Step	Hyp	Ref	Expression
1		ax-1cn 6977	. 2 ⊢ 1 ∈ ℂ
2		addcl 7006	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 401	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 1393  (class class class)co 5512  ℂcc 6887  1c1 6890   + caddc 6892

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 101  ax-1cn 6977  ax-addcl 6980

This theorem is referenced by:  nneo  8341  zeo  8343  zeo2  8344  zesq  9367