Theorem ex-id

Description: Example for df-id . Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015)

	Ref	Expression
Assertion	ex-id	$$⊢\left(5\mathrm{I}5\wedge \neg 4\mathrm{I}5\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢5=5$$
2		5re	$$⊢5\in ℝ$$
3	2	elexi	$$⊢5\in \mathrm{V}$$
4	3	ideq	$$⊢5\mathrm{I}5\leftrightarrow 5=5$$
5	1 4	mpbir	$$⊢5\mathrm{I}5$$
6		4re	$$⊢4\in ℝ$$
7		4lt5	$$⊢4<5$$
8	6 7	ltneii	$$⊢4\ne 5$$
9	3	ideq	$$⊢4\mathrm{I}5\leftrightarrow 4=5$$
10	8 9	nemtbir	$$⊢\neg 4\mathrm{I}5$$
11	5 10	pm3.2i	$$⊢\left(5\mathrm{I}5\wedge \neg 4\mathrm{I}5\right)$$