Theorem eq0al ≪ | index | src | ≫ 

theorem eq0al (A: set) {x: nat}: $ A == 0 <-> A. x ~x e. A $;
StepHypRefExpression
1 bitr 
(x e. A <-> x e. 0 <-> (x e. A <-> F.)) -> (x e. A <-> F. <-> ~x e. A) -> (x e. A <-> x e. 0 <-> ~x e. A)
2 bieq2 
(x e. 0 <-> F.) -> (x e. A <-> x e. 0 <-> (x e. A <-> F.))
3 eqfal 
x e. 0 <-> F. <-> ~x e. 0
4 el02 
~x e. 0
5 3, 4 mpbir 
x e. 0 <-> F.
6 2, 5 ax_mp 
x e. A <-> x e. 0 <-> (x e. A <-> F.)
7 1, 6 ax_mp 
(x e. A <-> F. <-> ~x e. A) -> (x e. A <-> x e. 0 <-> ~x e. A)
8 eqfal 
x e. A <-> F. <-> ~x e. A
9 7, 8 ax_mp 
x e. A <-> x e. 0 <-> ~x e. A
10 9 aleqi 
A. x (x e. A <-> x e. 0) <-> A. x ~x e. A
11 10 conv eqs 
A == 0 <-> A. x ~x e. 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)