Theorem 0cnd 7020

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 7019	. 2 ⊢ 0 ∈ ℂ
2	1	a1i 9	1 ⊢ (𝜑 → 0 ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 1393  ℂcc 6887  0cc0 6889

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-mulcl 6982  ax-i2m1 6989

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036

This theorem is referenced by:  mulap0r  7606  mulap0  7635  diveqap0  7661  eqneg  7708  prodgt0  7818  un0addcl  8215  un0mulcl  8216  iser0  9250  iser0f  9251  abs00ap  9660  abssubne0  9687  clim0c  9807