Theorem peano2cn 7243

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4336. (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 7069	. 2 ⊢ 1 ∈ ℂ
2		addcl 7098	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 415	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 1433  (class class class)co 5532  ℂcc 6979  1c1 6982   + caddc 6984

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 106  ax-1cn 7069  ax-addcl 7072

This theorem is referenced by:  xp1d2m1eqxm1d2  8283  nneo  8450  zeo  8452  zeo2  8453  zesq  9591  facndiv  9666  faclbnd  9668  faclbnd6  9671  odd2np1  10272