Theorem peano2cn 7890

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4504. (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 7706	. 2 ⊢ 1 ∈ ℂ
2		addcl 7738	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 421	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 1480  (class class class)co 5767  ℂcc 7611  1c1 7614   + caddc 7616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 107  ax-1cn 7706  ax-addcl 7709

This theorem is referenced by:  xp1d2m1eqxm1d2  8965  nneo  9147  zeo  9149  zeo2  9150  zesq  10403  facndiv  10478  faclbnd  10480  faclbnd6  10483  bcxmas  11251  trireciplem  11262  odd2np1  11559  abssinper  12916