Theorem elb0S ≪ | index | src | ≫ 

theorem elb0S (a b: nat): $ suc a e. b0 b <-> a e. b $;
StepHypRefExpression
1 bitr 
(suc a e. b0 b <-> 0 < suc a /\ suc a - 1 e. b) -> (0 < suc a /\ suc a - 1 e. b <-> a e. b) -> (suc a e. b0 b <-> a e. b)
2 elb0 
suc a e. b0 b <-> 0 < suc a /\ suc a - 1 e. b
3 1, 2 ax_mp 
(0 < suc a /\ suc a - 1 e. b <-> a e. b) -> (suc a e. b0 b <-> a e. b)
4 bitr 
(0 < suc a /\ suc a - 1 e. b <-> suc a - 1 e. b) -> (suc a - 1 e. b <-> a e. b) -> (0 < suc a /\ suc a - 1 e. b <-> a e. b)
5 bian1 
0 < suc a -> (0 < suc a /\ suc a - 1 e. b <-> suc a - 1 e. b)
6 lt01S 
0 < suc a
7 5, 6 ax_mp 
0 < suc a /\ suc a - 1 e. b <-> suc a - 1 e. b
8 4, 7 ax_mp 
(suc a - 1 e. b <-> a e. b) -> (0 < suc a /\ suc a - 1 e. b <-> a e. b)
9 eleq1 
suc a - 1 = a -> (suc a - 1 e. b <-> a e. b)
10 sucsub1 
suc a - 1 = a
11 9, 10 ax_mp 
suc a - 1 e. b <-> a e. b
12 8, 11 ax_mp 
0 < suc a /\ suc a - 1 e. b <-> a e. b
13 3, 12 ax_mp 
suc a e. b0 b <-> a e. b

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, muleq, add0, addS, mul0, mulS)