Theorem ssin ≪ | index | src | ≫ 

theorem ssin (A B C: set): $ A C_ B i^i C <-> A C_ B /\ A C_ C $;
StepHypRefExpression
1 inss1 
B i^i C C_ B
2 sstr 
A C_ B i^i C -> B i^i C C_ B -> A C_ B
3 1, 2 mpi 
A C_ B i^i C -> A C_ B
4 inss2 
B i^i C C_ C
5 sstr 
A C_ B i^i C -> B i^i C C_ C -> A C_ C
6 4, 5 mpi 
A C_ B i^i C -> A C_ C
7 3, 6 iand 
A C_ B i^i C -> A C_ B /\ A C_ C
8 elin 
x e. B i^i C <-> x e. B /\ x e. C
9 anll 
A C_ B /\ A C_ C /\ x e. A -> A C_ B
10 anr 
A C_ B /\ A C_ C /\ x e. A -> x e. A
11 9, 10 sseld 
A C_ B /\ A C_ C /\ x e. A -> x e. B
12 anlr 
A C_ B /\ A C_ C /\ x e. A -> A C_ C
13 12, 10 sseld 
A C_ B /\ A C_ C /\ x e. A -> x e. C
14 11, 13 iand 
A C_ B /\ A C_ C /\ x e. A -> x e. B /\ x e. C
15 8, 14 sylibr 
A C_ B /\ A C_ C /\ x e. A -> x e. B i^i C
16 15 exp 
A C_ B /\ A C_ C -> x e. A -> x e. B i^i C
17 16 iald 
A C_ B /\ A C_ C -> A. x (x e. A -> x e. B i^i C)
18 17 conv subset 
A C_ B /\ A C_ C -> A C_ B i^i C
19 7, 18 ibii 
A C_ B i^i C <-> A C_ B /\ A C_ C

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)