Theorem lmemS ≪ | index | src | ≫

theorem lmemS (a b l: nat): $ a IN b : l <-> a = b \/ a IN l $;

Step	Hyp	Ref	Expression
1		bitr	(a IN b : l <-> a e. b ; lmems l) -> (a e. b ; lmems l <-> a = b \/ a IN l) -> (a IN b : l <-> a = b \/ a IN l)
2		elneq2	lmems (b : l) = b ; lmems l -> (a e. lmems (b : l) <-> a e. b ; lmems l)
3	2	conv lmem	lmems (b : l) = b ; lmems l -> (a IN b : l <-> a e. b ; lmems l)
4		lmemsS	lmems (b : l) = b ; lmems l
5	3, 4	ax_mp	a IN b : l <-> a e. b ; lmems l
6	1, 5	ax_mp	(a e. b ; lmems l <-> a = b \/ a IN l) -> (a IN b : l <-> a = b \/ a IN l)
7		elins	a e. b ; lmems l <-> a = b \/ a e. lmems l
8	7	conv lmem	a e. b ; lmems l <-> a = b \/ a IN l
9	6, 8	ax_mp	a IN b : l <-> a = b \/ a IN l

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)