Theorem elin | index | src |

theorem elin (A B: set) (a: nat): $ a e. A i^i B <-> a e. A /\ a e. B $;
StepHypRefExpression
1 eleq1
x = a -> (x e. A <-> a e. A)
2 eleq1
x = a -> (x e. B <-> a e. B)
3 1, 2 aneqd
x = a -> (x e. A /\ x e. B <-> a e. A /\ a e. B)
4 3 elabe
a e. {x | x e. A /\ x e. B} <-> a e. A /\ a e. B
5 4 conv Inter
a e. A i^i B <-> a e. A /\ a e. 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)