Theorem ex2nal ≪ | index | src | ≫ 

theorem ex2nal (R: set) (l1 l2: nat):
  $ l1, l2 e. ex2 (Compl R) <-> len l1 = len l2 /\ ~l1, l2 e. all2 R $;
StepHypRefExpression
1 bitr 
(l1, l2 e. ex2 (Compl R) <-> len l1 = len l2 /\ ~l1, l2 e. all2 (Compl (Compl R))) ->
  (len l1 = len l2 /\ ~l1, l2 e. all2 (Compl (Compl R)) <-> len l1 = len l2 /\ ~l1, l2 e. all2 R) ->
  (l1, l2 e. ex2 (Compl R) <-> len l1 = len l2 /\ ~l1, l2 e. all2 R)
2 dfex2_2 
l1, l2 e. ex2 (Compl R) <-> len l1 = len l2 /\ ~l1, l2 e. all2 (Compl (Compl R))
3 1, 2 ax_mp 
(len l1 = len l2 /\ ~l1, l2 e. all2 (Compl (Compl R)) <-> len l1 = len l2 /\ ~l1, l2 e. all2 R) ->
  (l1, l2 e. ex2 (Compl R) <-> len l1 = len l2 /\ ~l1, l2 e. all2 R)
4 eleq2 
all2 (Compl (Compl R)) == all2 R -> (l1, l2 e. all2 (Compl (Compl R)) <-> l1, l2 e. all2 R)
5 all2eq 
Compl (Compl R) == R -> all2 (Compl (Compl R)) == all2 R
6 cplcpl 
Compl (Compl R) == R
7 5, 6 ax_mp 
all2 (Compl (Compl R)) == all2 R
8 4, 7 ax_mp 
l1, l2 e. all2 (Compl (Compl R)) <-> l1, l2 e. all2 R
9 8 noteqi 
~l1, l2 e. all2 (Compl (Compl R)) <-> ~l1, l2 e. all2 R
10 9 aneq2i 
len l1 = len l2 /\ ~l1, l2 e. all2 (Compl (Compl R)) <-> len l1 = len l2 /\ ~l1, l2 e. all2 R
11 3, 10 ax_mp 
l1, l2 e. ex2 (Compl R) <-> len l1 = len l2 /\ ~l1, l2 e. all2 R

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)