Theorem insss ≪ | index | src | ≫ 

theorem insss (A: set) (a b: nat): $ a ; b C_ A <-> a e. A /\ b C_ A $;
StepHypRefExpression
1 bitr 
(a ; b C_ A <-> sn a u. b C_ A) -> (sn a u. b C_ A <-> a e. A /\ b C_ A) -> (a ; b C_ A <-> a e. A /\ b C_ A)
2 sseq1 
a ; b == sn a u. b -> (a ; b C_ A <-> sn a u. b C_ A)
3 insunsn 
a ; b == sn a u. b
4 2, 3 ax_mp 
a ; b C_ A <-> sn a u. b C_ A
5 1, 4 ax_mp 
(sn a u. b C_ A <-> a e. A /\ b C_ A) -> (a ; b C_ A <-> a e. A /\ b C_ A)
6 bitr 
(sn a u. b C_ A <-> sn a C_ A /\ b C_ A) -> (sn a C_ A /\ b C_ A <-> a e. A /\ b C_ A) -> (sn a u. b C_ A <-> a e. A /\ b C_ A)
7 unss 
sn a u. b C_ A <-> sn a C_ A /\ b C_ A
8 6, 7 ax_mp 
(sn a C_ A /\ b C_ A <-> a e. A /\ b C_ A) -> (sn a u. b C_ A <-> a e. A /\ b C_ A)
9 snss 
sn a C_ A <-> a e. A
10 9 aneq1i 
sn a C_ A /\ b C_ A <-> a e. A /\ b C_ A
11 8, 10 ax_mp 
sn a u. b C_ A <-> a e. A /\ b C_ A
12 5, 11 ax_mp 
a ; b C_ A <-> a e. A /\ b C_ A

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)