Theorem zfstsnd0 ≪ | index | src | ≫ 

theorem zfstsnd0 (n: nat): $ zfst n = 0 \/ zsnd n = 0 $;
StepHypRefExpression
1 eor 
(n = b0 (n // 2) -> zfst n = 0 \/ zsnd n = 0) -> (n = b1 (n // 2) -> zfst n = 0 \/ zsnd n = 0) -> n = b0 (n // 2) \/ n = b1 (n // 2) -> zfst n = 0 \/ zsnd n = 0
2 zsndb0 
zsnd (b0 (n // 2)) = 0
3 zsndeq 
n = b0 (n // 2) -> zsnd n = zsnd (b0 (n // 2))
4 2, 3 syl6eq 
n = b0 (n // 2) -> zsnd n = 0
5 4 orrd 
n = b0 (n // 2) -> zfst n = 0 \/ zsnd n = 0
6 1, 5 ax_mp 
(n = b1 (n // 2) -> zfst n = 0 \/ zsnd n = 0) -> n = b0 (n // 2) \/ n = b1 (n // 2) -> zfst n = 0 \/ zsnd n = 0
7 zfstb1 
zfst (b1 (n // 2)) = 0
8 zfsteq 
n = b1 (n // 2) -> zfst n = zfst (b1 (n // 2))
9 7, 8 syl6eq 
n = b1 (n // 2) -> zfst n = 0
10 9 orld 
n = b1 (n // 2) -> zfst n = 0 \/ zsnd n = 0
11 6, 10 ax_mp 
n = b0 (n // 2) \/ n = b1 (n // 2) -> zfst n = 0 \/ zsnd n = 0
12 b0orb1 
n = b0 (n // 2) \/ n = b1 (n // 2)
13 11, 12 ax_mp 
zfst n = 0 \/ zsnd n = 0

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)