Theorem elopt ≪ | index | src | ≫

theorem elopt (A: set) (a: nat): $ a e. Option A <-> a = 0 \/ a - 1 e. A $;

Step	Hyp	Ref	Expression
1		eqeq1	n = a -> (n = 0 <-> a = 0)
2		subeq1	n = a -> n - 1 = a - 1
3	2	eleq1d	n = a -> (n - 1 e. A <-> a - 1 e. A)
4	1, 3	oreqd	n = a -> (n = 0 \/ n - 1 e. A <-> a = 0 \/ a - 1 e. A)
5	4	elabe	a e. {n \| n = 0 \/ n - 1 e. A} <-> a = 0 \/ a - 1 e. A
6	5	conv Option	a e. Option A <-> a = 0 \/ a - 1 e. 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), axs_the (theid, the0), axs_peano (addeq)