Theorem zleorle ≪ | index | src | ≫ 

theorem zleorle (a b: nat): $ a <=Z b \/ b <=Z a $;
StepHypRefExpression
1 oreq 
(zfst (a -Z b) = 0 <-> a <=Z b) -> (zsnd (a -Z b) = 0 <-> b <=Z a) -> (zfst (a -Z b) = 0 \/ zsnd (a -Z b) = 0 <-> a <=Z b \/ b <=Z a)
2 bitr 
(zfst (a -Z b) = 0 <-> a -Z b <=Z 0) -> (a -Z b <=Z 0 <-> a <=Z b) -> (zfst (a -Z b) = 0 <-> a <=Z b)
3 zfsteq0 
zfst (a -Z b) = 0 <-> a -Z b <=Z 0
4 2, 3 ax_mp 
(a -Z b <=Z 0 <-> a <=Z b) -> (zfst (a -Z b) = 0 <-> a <=Z b)
5 zlesub0 
a -Z b <=Z 0 <-> a <=Z b
6 4, 5 ax_mp 
zfst (a -Z b) = 0 <-> a <=Z b
7 1, 6 ax_mp 
(zsnd (a -Z b) = 0 <-> b <=Z a) -> (zfst (a -Z b) = 0 \/ zsnd (a -Z b) = 0 <-> a <=Z b \/ b <=Z a)
8 bitr 
(zsnd (a -Z b) = 0 <-> 0 <=Z a -Z b) -> (0 <=Z a -Z b <-> b <=Z a) -> (zsnd (a -Z b) = 0 <-> b <=Z a)
9 zsndeq0 
zsnd (a -Z b) = 0 <-> 0 <=Z a -Z b
10 8, 9 ax_mp 
(0 <=Z a -Z b <-> b <=Z a) -> (zsnd (a -Z b) = 0 <-> b <=Z a)
11 zle0sub 
0 <=Z a -Z b <-> b <=Z a
12 10, 11 ax_mp 
zsnd (a -Z b) = 0 <-> b <=Z a
13 7, 12 ax_mp 
zfst (a -Z b) = 0 \/ zsnd (a -Z b) = 0 <-> a <=Z b \/ b <=Z a
14 zfstsnd0 
zfst (a -Z b) = 0 \/ zsnd (a -Z b) = 0
15 13, 14 mpbi 
a <=Z b \/ b <=Z 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, muleq, add0, addS, mul0, mulS)