Theorem peano2cn 8095

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4596. (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 7907	. 2 ⊢ 1 ∈ ℂ
2		addcl 7939	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 425	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 2148  (class class class)co 5878  ℂcc 7812  1c1 7815   + caddc 7817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-1cn 7907  ax-addcl 7910

This theorem is referenced by:  xp1d2m1eqxm1d2  9174  nneo  9359  zeo  9361  zeo2  9362  zesq  10642  facndiv  10722  faclbnd  10724  faclbnd6  10727  bcxmas  11500  trireciplem  11511  odd2np1  11881  abssinper  14407  lgseisenlem1  14590