Theorem peano2cn 8178

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 7989	. 2 ⊢ 1 ∈ ℂ
2		addcl 8021	. 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 7894  1c1 7897   + caddc 7899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-1cn 7989  ax-addcl 7992

This theorem is referenced by:  xp1d2m1eqxm1d2  9261  nneo  9446  zeo  9448  zeo2  9449  zesq  10767  facndiv  10848  faclbnd  10850  faclbnd6  10853  bcxmas  11671  trireciplem  11682  odd2np1  12055  abssinper  15166  lgseisenlem1  15395  lgsquadlem1  15402