Theorem peano2cn 8407

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4716. (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 8219	. 2 ⊢ 1 ∈ ℂ
2		addcl 8251	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 425	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 2203  (class class class)co 6049  ℂcc 8124  1c1 8127   + caddc 8129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-1cn 8219  ax-addcl 8222

This theorem is referenced by:  xp1d2m1eqxm1d2  9490  nneo  9680  zeo  9682  zeo2  9683  zesq  11019  facndiv  11100  faclbnd  11102  faclbnd6  11105  bcxmas  12171  trireciplem  12182  odd2np1  12555  abssinper  15703  lgseisenlem1  15935  lgsquadlem1  15942