Theorem sscpl ≪ | index | src | ≫ 

theorem sscpl (A B: set): $ B C_ A <-> Compl A C_ Compl B $;
StepHypRefExpression
1 bitr4 
(x e. B -> x e. A <-> ~x e. A -> ~x e. B) -> (x e. Compl A -> x e. Compl B <-> ~x e. A -> ~x e. B) -> (x e. B -> x e. A <-> x e. Compl A -> x e. Compl B)
2 con3bi 
x e. B -> x e. A <-> ~x e. A -> ~x e. B
3 1, 2 ax_mp 
(x e. Compl A -> x e. Compl B <-> ~x e. A -> ~x e. B) -> (x e. B -> x e. A <-> x e. Compl A -> x e. Compl B)
4 elcpl 
x e. Compl A <-> ~x e. A
5 elcpl 
x e. Compl B <-> ~x e. B
6 4, 5 imeqi 
x e. Compl A -> x e. Compl B <-> ~x e. A -> ~x e. B
7 3, 6 ax_mp 
x e. B -> x e. A <-> x e. Compl A -> x e. Compl B
8 7 aleqi 
A. x (x e. B -> x e. A) <-> A. x (x e. Compl A -> x e. Compl B)
9 8 conv subset 
B C_ A <-> Compl A C_ Compl B

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)