Theorem ellower ≪ | index | src | ≫

theorem ellower (A: set) (a: nat): $ finite A -> (a e. lower A <-> a e. A) $;


(finite A <-> A == lower A) -> finite A -> A == lower A
finite A <-> A == lower A
finite A -> A == lower A
finite A -> (a e. A <-> a e. lower A)
finite A -> (a e. lower A <-> a e. A)

Step	Hyp	Ref	Expression
1		bi1	(finite A <-> A == lower A) -> finite A -> A == lower A
2		eqlower	finite A <-> A == lower A
3	1, 2	ax_mp	finite A -> A == lower A
4	3	eleq2d	finite A -> (a e. A <-> a e. lower A)
5	4	bicomd	finite A -> (a e. lower A <-> a e. 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)