Theorem ltfin ≪ | index | src | ≫

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

Step	Hyp	Ref	Expression
1		lteq2	m = n -> (y < m <-> y < n)
2	1	imeq2d	m = n -> (y e. {x \| x < n} -> y < m <-> y e. {x \| x < n} -> y < n)
3	2	aleqd	m = n -> (A. y (y e. {x \| x < n} -> y < m) <-> A. y (y e. {x \| x < n} -> y < n))
4	3	iexe	A. y (y e. {x \| x < n} -> y < n) -> E. m A. y (y e. {x \| x < n} -> y < m)
5	4	conv finite	A. y (y e. {x \| x < n} -> y < n) -> finite {x \| x < n}
6		lteq1	x = y -> (x < n <-> y < n)
7	6	elabe	y e. {x \| x < n} <-> y < n
8	7	bi1i	y e. {x \| x < n} -> y < n
9	8	ax_gen	A. y (y e. {x \| x < n} -> y < n)
10	5, 9	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, addeq)