Theorem uncpl2 ≪ | index | src | ≫

theorem uncpl2 (A: set): $ A u. Compl A == _V $;

Step	Hyp	Ref	Expression
1		bith	x e. A u. Compl A -> x e. _V -> (x e. A u. Compl A <-> x e. _V)
2		elun	x e. A u. Compl A <-> x e. A \/ x e. Compl A
3		elcpl	x e. Compl A <-> ~x e. A
4	3	oreq2i	x e. A \/ x e. Compl A <-> x e. A \/ ~x e. A
5		em	x e. A \/ ~x e. A
6	4, 5	mpbir	x e. A \/ x e. Compl A
7	2, 6	mpbir	x e. A u. Compl A
8	1, 7	ax_mp	x e. _V -> (x e. A u. Compl A <-> x e. _V)
9		elv	x e. _V
10	8, 9	ax_mp	x e. A u. Compl A <-> x e. _V
11	10	eqri	A u. Compl A == _V

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)