Theorem sizess ≪ | index | src | ≫

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


A C_ upto k -> size A <= k
finite A -> A C_ upto k -> size A <= k
upto (size A) C_ upto k <-> size A <= k
finite A <-> A C_ upto (size A)
A C_ upto (size A) -> upto (size A) C_ upto k -> A C_ upto k
finite A -> upto (size A) C_ upto k -> A C_ upto k
finite A -> size A <= k -> A C_ upto k
finite A -> (A C_ upto k <-> size A <= k)

Step	Hyp	Ref	Expression
1		sizess1	A C_ upto k -> size A <= k
2	1	a1i	finite A -> A C_ upto k -> size A <= k
3		uptoss	upto (size A) C_ upto k <-> size A <= k
4		sssize	finite A <-> A C_ upto (size A)
5		sstr	A C_ upto (size A) -> upto (size A) C_ upto k -> A C_ upto k
6	4, 5	sylbi	finite A -> upto (size A) C_ upto k -> A C_ upto k
7	3, 6	syl5bir	finite A -> size A <= k -> A C_ upto k
8	2, 7	ibid	finite A -> (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, mul0, mulS)