Theorem peano2cn 8297

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4688. (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 8108	. 2 ⊢ 1 ∈ ℂ
2		addcl 8140	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 425	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 2200  (class class class)co 6010  ℂcc 8013  1c1 8016   + caddc 8018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-1cn 8108  ax-addcl 8111

This theorem is referenced by:  xp1d2m1eqxm1d2  9380  nneo  9566  zeo  9568  zeo2  9569  zesq  10897  facndiv  10978  faclbnd  10980  faclbnd6  10983  bcxmas  12021  trireciplem  12032  odd2np1  12405  abssinper  15541  lgseisenlem1  15770  lgsquadlem1  15777