Theorem peano2cn 7814

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4467. (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 7632	. 2 ⊢ 1 ∈ ℂ
2		addcl 7663	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 419	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 1461  (class class class)co 5726  ℂcc 7539  1c1 7542   + caddc 7544

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 107  ax-1cn 7632  ax-addcl 7635

This theorem is referenced by:  xp1d2m1eqxm1d2  8870  nneo  9052  zeo  9054  zeo2  9055  zesq  10297  facndiv  10372  faclbnd  10374  faclbnd6  10377  bcxmas  11144  trireciplem  11155  odd2np1  11412