Theorem eqlower1 ≪ | index | src | ≫

theorem eqlower1 (A: set) (a: nat): $ finite A -> (A == a <-> lower A = a) $;


lower a = a
A == a -> lower A = lower a
A == a -> lower A = a
finite A /\ A == a -> lower A = a
finite A <-> A == lower A
finite A /\ lower A = a -> finite A
finite A /\ lower A = a -> A == lower A
finite A /\ lower A = a -> lower A = a
finite A /\ lower A = a -> lower A == a
finite A /\ lower A = a -> A == a
finite A -> (A == a <-> lower A = a)

Step	Hyp	Ref	Expression
1		lowerns	lower a = a
2		lowereq	A == a -> lower A = lower a
3	1, 2	syl6eq	A == a -> lower A = a
4	3	anwr	finite A /\ A == a -> lower A = a
5		eqlower	finite A <-> A == lower A
6		anl	finite A /\ lower A = a -> finite A
7	5, 6	sylib	finite A /\ lower A = a -> A == lower A
8		anr	finite A /\ lower A = a -> lower A = a
9	8	nseqd	finite A /\ lower A = a -> lower A == a
10	7, 9	eqstrd	finite A /\ lower A = a -> A == a
11	4, 10	ibida	finite A -> (A == a <-> lower A = 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, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)