Theorem peano2cn 8156

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4628. (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 7967	. 2 ⊢ 1 ∈ ℂ
2		addcl 7999	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 425	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 2164  (class class class)co 5919  ℂcc 7872  1c1 7875   + caddc 7877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-1cn 7967  ax-addcl 7970

This theorem is referenced by:  xp1d2m1eqxm1d2  9238  nneo  9423  zeo  9425  zeo2  9426  zesq  10732  facndiv  10813  faclbnd  10815  faclbnd6  10818  bcxmas  11635  trireciplem  11646  odd2np1  12017  abssinper  15022  lgseisenlem1  15227  lgsquadlem1  15234