Theorem elunii ≪ | index | src | ≫ 

theorem elunii (A: set) (a b: nat): $ a e. A -> b e. a -> b e. sUnion A $;
StepHypRefExpression
1 eluni 
b e. sUnion A <-> E. a1 (b e. a1 /\ a1 e. A)
2 elneq2 
a1 = a -> (b e. a1 <-> b e. a)
3 eleq1 
a1 = a -> (a1 e. A <-> a e. A)
4 2, 3 aneqd 
a1 = a -> (b e. a1 /\ a1 e. A <-> b e. a /\ a e. A)
5 4 iexe 
b e. a /\ a e. A -> E. a1 (b e. a1 /\ a1 e. A)
6 1, 5 sylibr 
b e. a /\ a e. A -> b e. sUnion A
7 6 expcom 
a e. A -> b e. a -> b e. sUnion 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 (peano2, addeq, muleq)