Theorem 0cnd 7112

Description: 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)

Assertion

Ref	Expression
0cnd	⊢ (𝜑 → 0 ∈ ℂ)

Proof of Theorem 0cnd

Step	Hyp	Ref	Expression
1		0cn 7111	. 2 ⊢ 0 ∈ ℂ
2	1	a1i 9	1 ⊢ (𝜑 → 0 ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 1433  ℂcc 6979  0cc0 6981

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-mulcl 7074  ax-i2m1 7081

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by:  mulap0r  7715  mulap0  7744  diveqap0  7770  eqneg  7820  prodgt0  7930  un0addcl  8321  un0mulcl  8322  modsumfzodifsn  9398  iser0  9471  iser0f  9472  abs00ap  9948  abssubne0  9977  clim0c  10125