Theorem snss ≪ | index | src | ≫

theorem snss (A: set) (a: nat): $ sn a C_ A <-> a e. A $;

Step	Hyp	Ref	Expression
1		bitr	(sn a C_ A <-> A. x (x = a -> x e. A)) -> (A. x (x = a -> x e. A) <-> a e. A) -> (sn a C_ A <-> a e. A)
2		elsn	x e. sn a <-> x = a
3	2	imeq1i	x e. sn a -> x e. A <-> x = a -> x e. A
4	3	aleqi	A. x (x e. sn a -> x e. A) <-> A. x (x = a -> x e. A)
5	4	conv subset	sn a C_ A <-> A. x (x = a -> x e. A)
6	1, 5	ax_mp	(A. x (x = a -> x e. A) <-> a e. A) -> (sn a C_ A <-> a e. A)
7		eleq1	x = a -> (x e. A <-> a e. A)
8	7	aleqe	A. x (x = a -> x e. A) <-> a e. A
9	6, 8	ax_mp	sn a C_ A <-> a e. 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)