Theorem lowerinj ≪ | index | src | ≫

theorem lowerinj (A B: set):
  $ finite A -> finite B -> (A == B <-> lower A = lower B) $;


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

Step	Hyp	Ref	Expression
1		eqlower	finite A <-> A == lower A
2		anl	finite A /\ finite B -> finite A
3	1, 2	sylib	finite A /\ finite B -> A == lower A
4	3	eqseq1d	finite A /\ finite B -> (A == B <-> lower A == B)
5		eqlower2	finite B -> (lower A == B <-> lower A = lower B)
6	5	anwr	finite A /\ finite B -> (lower A == B <-> lower A = lower B)
7	4, 6	bitrd	finite A /\ finite B -> (A == B <-> lower A = lower B)
8	7	exp	finite A -> finite B -> (A == B <-> lower A = lower B)

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)