Theorem b1ins ≪ | index | src | ≫ 

theorem b1ins (n: nat): $ b1 n = 0 ; b0 n $;
StepHypRefExpression
1 axext 
b1 n == 0 ; b0 n -> b1 n = 0 ; b0 n
2 elb1 
x e. b1 n <-> x = 0 \/ x - 1 e. n
3 bitr 
(x e. 0 ; b0 n <-> x = 0 \/ x e. b0 n) -> (x = 0 \/ x e. b0 n <-> x = 0 \/ 0 < x /\ x - 1 e. n) -> (x e. 0 ; b0 n <-> x = 0 \/ 0 < x /\ x - 1 e. n)
4 elins 
x e. 0 ; b0 n <-> x = 0 \/ x e. b0 n
5 3, 4 ax_mp 
(x = 0 \/ x e. b0 n <-> x = 0 \/ 0 < x /\ x - 1 e. n) -> (x e. 0 ; b0 n <-> x = 0 \/ 0 < x /\ x - 1 e. n)
6 oreq2 
(x e. b0 n <-> 0 < x /\ x - 1 e. n) -> (x = 0 \/ x e. b0 n <-> x = 0 \/ 0 < x /\ x - 1 e. n)
7 elb0 
x e. b0 n <-> 0 < x /\ x - 1 e. n
8 6, 7 ax_mp 
x = 0 \/ x e. b0 n <-> x = 0 \/ 0 < x /\ x - 1 e. n
9 5, 8 ax_mp 
x e. 0 ; b0 n <-> x = 0 \/ 0 < x /\ x - 1 e. n
10 orl 
x = 0 -> x = 0 \/ x - 1 e. n
11 orl 
x = 0 -> x = 0 \/ 0 < x /\ x - 1 e. n
12 10, 11 bithd 
x = 0 -> (x = 0 \/ x - 1 e. n <-> x = 0 \/ 0 < x /\ x - 1 e. n)
13 lt01 
0 < x <-> x != 0
14 13 conv ne 
0 < x <-> ~x = 0
15 bian1 
0 < x -> (0 < x /\ x - 1 e. n <-> x - 1 e. n)
16 14, 15 sylbir 
~x = 0 -> (0 < x /\ x - 1 e. n <-> x - 1 e. n)
17 16 bicomd 
~x = 0 -> (x - 1 e. n <-> 0 < x /\ x - 1 e. n)
18 17 oreq2d 
~x = 0 -> (x = 0 \/ x - 1 e. n <-> x = 0 \/ 0 < x /\ x - 1 e. n)
19 12, 18 cases 
x = 0 \/ x - 1 e. n <-> x = 0 \/ 0 < x /\ x - 1 e. n
20 2, 9, 19 bitr4gi 
x e. b1 n <-> x e. 0 ; b0 n
21 20 ax_gen 
A. x (x e. b1 n <-> x e. 0 ; b0 n)
22 21 conv eqs 
b1 n == 0 ; b0 n
23 1, 22 ax_mp 
b1 n = 0 ; b0 n

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)