Theorem 0re

Description: The number 0 is real. Remark: the first step could also be ax-icn . See also 0reALT . (Contributed by Eric Schmidt, 21-May-2007) (Revised by Scott Fenton, 3-Jan-2013) Reduce dependencies on axioms. (Revised by Steven Nguyen, 11-Oct-2022)

	Ref	Expression
Assertion	0re	$$⊢0\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	$$⊢1\in ℂ$$
2		cnre	$$⊢1\in ℂ\to \exists x\in ℝ\exists y\in ℝ1=x+\mathrm{i}y$$
3		ax-rnegex	$$⊢x\in ℝ\to \exists z\in ℝx+z=0$$
4		readdcl	$$⊢\left(x\in ℝ\wedge z\in ℝ\right)\to x+z\in ℝ$$
5		eleq1	$$⊢x+z=0\to \left(x+z\in ℝ\leftrightarrow 0\in ℝ\right)$$
6	4 5	syl5ibcom	$$⊢\left(x\in ℝ\wedge z\in ℝ\right)\to \left(x+z=0\to 0\in ℝ\right)$$
7	6	rexlimdva	$$⊢x\in ℝ\to \left(\exists z\in ℝx+z=0\to 0\in ℝ\right)$$
8	3 7	mpd	$$⊢x\in ℝ\to 0\in ℝ$$
9	8	adantr	$$⊢\left(x\in ℝ\wedge \exists y\in ℝ1=x+\mathrm{i}y\right)\to 0\in ℝ$$
10	9	rexlimiva	$$⊢\exists x\in ℝ\exists y\in ℝ1=x+\mathrm{i}y\to 0\in ℝ$$
11	1 2 10	mp2b	$$⊢0\in ℝ$$