Theorem elBool ≪ | index | src | ≫ 

theorem elBool (n: nat): $ n e. Bool <-> bool n $;
StepHypRefExpression
1 bitr 
(n e. Bool <-> n = 0 \/ n e. sn 1) -> (n = 0 \/ n e. sn 1 <-> bool n) -> (n e. Bool <-> bool n)
2 elins 
n e. 0 ; sn 1 <-> n = 0 \/ n e. sn 1
3 2 conv Bool 
n e. Bool <-> n = 0 \/ n e. sn 1
4 1, 3 ax_mp 
(n = 0 \/ n e. sn 1 <-> bool n) -> (n e. Bool <-> bool n)
5 bitr4 
(n = 0 \/ n e. sn 1 <-> n = 0 \/ n = 1) -> (bool n <-> n = 0 \/ n = 1) -> (n = 0 \/ n e. sn 1 <-> bool n)
6 elsn 
n e. sn 1 <-> n = 1
7 6 oreq2i 
n = 0 \/ n e. sn 1 <-> n = 0 \/ n = 1
8 5, 7 ax_mp 
(bool n <-> n = 0 \/ n = 1) -> (n = 0 \/ n e. sn 1 <-> bool n)
9 bool01 
bool n <-> n = 0 \/ n = 1
10 8, 9 ax_mp 
n = 0 \/ n e. sn 1 <-> bool n
11 4, 10 ax_mp 
n e. Bool <-> bool 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)