Theorem peano2cn 8180

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4632. (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 7991	. 2 ⊢ 1 ∈ ℂ
2		addcl 8023	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 425	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 2167  (class class class)co 5925  ℂcc 7896  1c1 7899   + caddc 7901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-1cn 7991  ax-addcl 7994

This theorem is referenced by:  xp1d2m1eqxm1d2  9263  nneo  9448  zeo  9450  zeo2  9451  zesq  10769  facndiv  10850  faclbnd  10852  faclbnd6  10855  bcxmas  11673  trireciplem  11684  odd2np1  12057  abssinper  15190  lgseisenlem1  15419  lgsquadlem1  15426