Theorem uneq0 ≪ | index | src | ≫ 

theorem uneq0 (A B: set): $ A u. B == 0 <-> A == 0 /\ B == 0 $;
StepHypRefExpression
1 bitr3 
(A u. B C_ 0 <-> A u. B == 0) -> (A u. B C_ 0 <-> A == 0 /\ B == 0) -> (A u. B == 0 <-> A == 0 /\ B == 0)
2 ss02 
A u. B C_ 0 <-> A u. B == 0
3 1, 2 ax_mp 
(A u. B C_ 0 <-> A == 0 /\ B == 0) -> (A u. B == 0 <-> A == 0 /\ B == 0)
4 bitr 
(A u. B C_ 0 <-> A C_ 0 /\ B C_ 0) -> (A C_ 0 /\ B C_ 0 <-> A == 0 /\ B == 0) -> (A u. B C_ 0 <-> A == 0 /\ B == 0)
5 unss 
A u. B C_ 0 <-> A C_ 0 /\ B C_ 0
6 4, 5 ax_mp 
(A C_ 0 /\ B C_ 0 <-> A == 0 /\ B == 0) -> (A u. B C_ 0 <-> A == 0 /\ B == 0)
7 aneq 
(A C_ 0 <-> A == 0) -> (B C_ 0 <-> B == 0) -> (A C_ 0 /\ B C_ 0 <-> A == 0 /\ B == 0)
8 ss02 
A C_ 0 <-> A == 0
9 7, 8 ax_mp 
(B C_ 0 <-> B == 0) -> (A C_ 0 /\ B C_ 0 <-> A == 0 /\ B == 0)
10 ss02 
B C_ 0 <-> B == 0
11 9, 10 ax_mp 
A C_ 0 /\ B C_ 0 <-> A == 0 /\ B == 0
12 6, 11 ax_mp 
A u. B C_ 0 <-> A == 0 /\ B == 0
13 3, 12 ax_mp 
A u. B == 0 <-> A == 0 /\ B == 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)