Theorem optS ≪ | index | src | ≫ 

theorem optS (A: set) (a: nat): $ suc a e. Option A <-> a e. A $;
StepHypRefExpression
1 bitr 
(suc a e. Option A <-> suc a = 0 \/ suc a - 1 e. A) -> (suc a = 0 \/ suc a - 1 e. A <-> a e. A) -> (suc a e. Option A <-> a e. A)
2 elopt 
suc a e. Option A <-> suc a = 0 \/ suc a - 1 e. A
3 1, 2 ax_mp 
(suc a = 0 \/ suc a - 1 e. A <-> a e. A) -> (suc a e. Option A <-> a e. A)
4 bitr 
(suc a = 0 \/ suc a - 1 e. A <-> suc a - 1 e. A) -> (suc a - 1 e. A <-> a e. A) -> (suc a = 0 \/ suc a - 1 e. A <-> a e. A)
5 bior1 
~suc a = 0 -> (suc a = 0 \/ suc a - 1 e. A <-> suc a - 1 e. A)
6 peano1 
suc a != 0
7 6 conv ne 
~suc a = 0
8 5, 7 ax_mp 
suc a = 0 \/ suc a - 1 e. A <-> suc a - 1 e. A
9 4, 8 ax_mp 
(suc a - 1 e. A <-> a e. A) -> (suc a = 0 \/ suc a - 1 e. A <-> a e. A)
10 eleq1 
suc a - 1 = a -> (suc a - 1 e. A <-> a e. A)
11 sucsub1 
suc a - 1 = a
12 10, 11 ax_mp 
suc a - 1 e. A <-> a e. A
13 9, 12 ax_mp 
suc a = 0 \/ suc a - 1 e. A <-> a e. A
14 3, 13 ax_mp 
suc a e. Option A <-> a 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 (peano1, peano2, peano5, addeq, add0, addS)