Theorem ssins ≪ | index | src | ≫ 

theorem ssins (A: set) (a b: nat): $ A C_ a ; b -> a e. A \/ A C_ b $;
StepHypRefExpression
1 imeq2a 
(x e. A -> (x e. a ; b <-> x e. b)) -> (x e. A -> x e. a ; b <-> x e. A -> x e. b)
2 elins 
x e. a ; b <-> x = a \/ x e. b
3 bior1 
~x = a -> (x = a \/ x e. b <-> x e. b)
4 eleq1 
x = a -> (x e. A <-> a e. A)
5 4 bi1d 
x = a -> x e. A -> a e. A
6 5 com12 
x e. A -> x = a -> a e. A
7 6 con3d 
x e. A -> ~a e. A -> ~x = a
8 7 impcom 
~a e. A /\ x e. A -> ~x = a
9 3, 8 syl 
~a e. A /\ x e. A -> (x = a \/ x e. b <-> x e. b)
10 2, 9 syl5bb 
~a e. A /\ x e. A -> (x e. a ; b <-> x e. b)
11 10 exp 
~a e. A -> x e. A -> (x e. a ; b <-> x e. b)
12 1, 11 syl 
~a e. A -> (x e. A -> x e. a ; b <-> x e. A -> x e. b)
13 12 aleqd 
~a e. A -> (A. x (x e. A -> x e. a ; b) <-> A. x (x e. A -> x e. b))
14 13 conv subset 
~a e. A -> (A C_ a ; b <-> A C_ b)
15 14 bi1d 
~a e. A -> A C_ a ; b -> A C_ b
16 15 com12 
A C_ a ; b -> ~a e. A -> A C_ b
17 16 conv or 
A C_ a ; b -> a e. A \/ A C_ 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), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)