Theorem lefin ≪ | index | src | ≫

theorem lefin (n: nat) {x: nat}: $ finite {x | x <= n} $;

Step	Hyp	Ref	Expression
1		lteq2	m = suc n -> (y < m <-> y < suc n)
2	1	imeq2d	m = suc n -> (y e. {x \| x <= n} -> y < m <-> y e. {x \| x <= n} -> y < suc n)
3	2	aleqd	m = suc n -> (A. y (y e. {x \| x <= n} -> y < m) <-> A. y (y e. {x \| x <= n} -> y < suc n))
4	3	iexe	A. y (y e. {x \| x <= n} -> y < suc n) -> E. m A. y (y e. {x \| x <= n} -> y < m)
5	4	conv finite	A. y (y e. {x \| x <= n} -> y < suc n) -> finite {x \| x <= n}
6		leeq1	x = y -> (x <= n <-> y <= n)
7	6	elabe	y e. {x \| x <= n} <-> y <= n
8		leltsuc	y <= n <-> y < suc n
9	8	bi1i	y <= n -> y < suc n
10	7, 9	sylbi	y e. {x \| x <= n} -> y < suc n
11	10	ax_gen	A. y (y e. {x \| x <= n} -> y < suc n)
12	5, 11	ax_mp	finite {x \| x <= n}

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_peano (peano2, peano5, addeq, add0, addS)