Theorem peano2cn 8033

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4572. (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 7846	. 2 ⊢ 1 ∈ ℂ
2		addcl 7878	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 422	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 2136  (class class class)co 5842  ℂcc 7751  1c1 7754   + caddc 7756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 107  ax-1cn 7846  ax-addcl 7849

This theorem is referenced by:  xp1d2m1eqxm1d2  9109  nneo  9294  zeo  9296  zeo2  9297  zesq  10573  facndiv  10652  faclbnd  10654  faclbnd6  10657  bcxmas  11430  trireciplem  11441  odd2np1  11810  abssinper  13407