Theorem sizess1 ≪ | index | src | ≫

theorem sizess1 (A: set) (k: nat): $ A C_ upto k -> size A <= k $;

Step	Hyp	Ref	Expression
1		uptoeq	x = k -> upto x = upto k
2	1	nseqd	x = k -> upto x == upto k
3	2	sseq2d	x = k -> (A C_ upto x <-> A C_ upto k)
4	3	elabe	k e. {x \| A C_ upto x} <-> A C_ upto k
5		leastle	k e. {x \| A C_ upto x} -> least {x \| A C_ upto x} <= k
6	5	conv size	k e. {x \| A C_ upto x} -> size A <= k
7	4, 6	sylbir	A C_ upto k -> size A <= k

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)