Theorem Arrowfin ≪ | index | src | ≫

theorem Arrowfin (A B: set): $ finite A -> finite B -> finite (Arrow A B) $;


Arrow A B C_ Power (Xp A B) -> finite (Power (Xp A B)) -> finite (Arrow A B)
Arrow A B C_ Power (Xp A B)
finite (Power (Xp A B)) -> finite (Arrow A B)
finite (Xp A B) -> finite (Power (Xp A B))
finite A -> finite B -> finite (Xp A B)
finite A /\ finite B -> finite (Xp A B)
finite A /\ finite B -> finite (Power (Xp A B))
finite A /\ finite B -> finite (Arrow A B)
finite A -> finite B -> finite (Arrow A B)

Step	Hyp	Ref	Expression
1		finss	Arrow A B C_ Power (Xp A B) -> finite (Power (Xp A B)) -> finite (Arrow A B)
2		Arrowssxp	Arrow A B C_ Power (Xp A B)
3	2	finss*	finite (Power (Xp A B)) -> finite (Arrow A B)
4		powerfin	finite (Xp A B) -> finite (Power (Xp A B))
5		xpfin	finite A -> finite B -> finite (Xp A B)
6	5	imp	finite A /\ finite B -> finite (Xp A B)
7	4, 6	syl	finite A /\ finite B -> finite (Power (Xp A B))
8	3, 7	syl	finite A /\ finite B -> finite (Arrow A B)
9	8	exp	finite A -> finite B -> finite (Arrow A 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)