Theorem subsnfin ≪ | index | src | ≫

theorem subsnfin (A: set): $ subsn A -> finite A $;

Step	Hyp	Ref	Expression
1		snfin	finite {a1 \| a1 = least A}
2		finss	A C_ {a1 \| a1 = least A} -> finite {a1 \| a1 = least A} -> finite A
3	1, 2	mpi	A C_ {a1 \| a1 = least A} -> finite A
4		ssab2	A. a1 (a1 e. A -> a1 = least A) <-> A C_ {a1 \| a1 = least A}
5		anl	subsn A /\ a1 e. A -> subsn A
6		anr	subsn A /\ a1 e. A -> a1 e. A
7		leastel	a1 e. A -> least A e. A
8	7	anwr	subsn A /\ a1 e. A -> least A e. A
9	5, 6, 8	subsni	subsn A /\ a1 e. A -> a1 = least A
10	9	ialda	subsn A -> A. a1 (a1 e. A -> a1 = least A)
11	4, 10	sylib	subsn A -> A C_ {a1 \| a1 = least A}
12	3, 11	syl	subsn A -> finite 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), axs_peano (peano1, peano2, peano5, addeq, add0, addS)