Theorem xnegex

Description: A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xnegex	$$⊢-A\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		df-xneg	$$⊢-A=if\left(A=\mathrm{+\infty },\mathrm{-\infty },if\left(A=\mathrm{-\infty },\mathrm{+\infty },-A\right)\right)$$
2		mnfxr	$$⊢\mathrm{-\infty }\in {ℝ}^{*}$$
3	2	elexi	$$⊢\mathrm{-\infty }\in \mathrm{V}$$
4		pnfex	$$⊢\mathrm{+\infty }\in \mathrm{V}$$
5		negex	$$⊢-A\in \mathrm{V}$$
6	4 5	ifex	$$⊢if\left(A=\mathrm{-\infty },\mathrm{+\infty },-A\right)\in \mathrm{V}$$
7	3 6	ifex	$$⊢if\left(A=\mathrm{+\infty },\mathrm{-\infty },if\left(A=\mathrm{-\infty },\mathrm{+\infty },-A\right)\right)\in \mathrm{V}$$
8	1 7	eqeltri	$$⊢-A\in \mathrm{V}$$