Theorem lmem1 ≪ | index | src | ≫

theorem lmem1 (a b: nat): $ a IN b : 0 <-> a = b $;

Step	Hyp	Ref	Expression
1		bitr	(a IN b : 0 <-> a = b \/ a IN 0) -> (a = b \/ a IN 0 <-> a = b) -> (a IN b : 0 <-> a = b)
2		lmemS	a IN b : 0 <-> a = b \/ a IN 0
3	1, 2	ax_mp	(a = b \/ a IN 0 <-> a = b) -> (a IN b : 0 <-> a = b)
4		bior2	~a IN 0 -> (a = b \/ a IN 0 <-> a = b)
5		lmem0	~a IN 0
6	4, 5	ax_mp	a = b \/ a IN 0 <-> a = b
7	3, 6	ax_mp	a IN b : 0 <-> a = 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)