Theorem sndisj ≪ | index | src | ≫

theorem sndisj (A: set) (a: nat): $ sn a i^i A == 0 <-> ~a e. A $;

Step	Hyp	Ref	Expression
1		bitr	(sn a i^i A == 0 <-> sn a C_ Compl A) -> (sn a C_ Compl A <-> ~a e. A) -> (sn a i^i A == 0 <-> ~a e. A)
2		ineq0	sn a i^i A == 0 <-> sn a C_ Compl A
3	1, 2	ax_mp	(sn a C_ Compl A <-> ~a e. A) -> (sn a i^i A == 0 <-> ~a e. A)
4		bitr	(sn a C_ Compl A <-> a e. Compl A) -> (a e. Compl A <-> ~a e. A) -> (sn a C_ Compl A <-> ~a e. A)
5		snss	sn a C_ Compl A <-> a e. Compl A
6	4, 5	ax_mp	(a e. Compl A <-> ~a e. A) -> (sn a C_ Compl A <-> ~a e. A)
7		elcpl	a e. Compl A <-> ~a e. A
8	6, 7	ax_mp	sn a C_ Compl A <-> ~a e. A
9	3, 8	ax_mp	sn a i^i A == 0 <-> ~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)