Theorem cplcpl ≪ | index | src | ≫

theorem cplcpl (A: set): $ Compl (Compl A) == A $;

Step	Hyp	Ref	Expression
1		bitr	(x e. Compl (Compl A) <-> ~x e. Compl A) -> (~x e. Compl A <-> x e. A) -> (x e. Compl (Compl A) <-> x e. A)
2		elcpl	x e. Compl (Compl A) <-> ~x e. Compl A
3	1, 2	ax_mp	(~x e. Compl A <-> x e. A) -> (x e. Compl (Compl A) <-> x e. A)
4		bitr4	(~x e. Compl A <-> ~~x e. A) -> (x e. A <-> ~~x e. A) -> (~x e. Compl A <-> x e. A)
5		noteq	(x e. Compl A <-> ~x e. A) -> (~x e. Compl A <-> ~~x e. A)
6		elcpl	x e. Compl A <-> ~x e. A
7	5, 6	ax_mp	~x e. Compl A <-> ~~x e. A
8	4, 7	ax_mp	(x e. A <-> ~~x e. A) -> (~x e. Compl A <-> x e. A)
9		notnot	x e. A <-> ~~x e. A
10	8, 9	ax_mp	~x e. Compl A <-> x e. A
11	3, 10	ax_mp	x e. Compl (Compl A) <-> x e. A
12	11	eqri	Compl (Compl A) == A

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8)