Theorem ineq0r ≪ | index | src | ≫

theorem ineq0r (A B: set): $ A i^i B == 0 <-> B C_ Compl A $;

Step	Hyp	Ref	Expression
1		bitr	(A i^i B == 0 <-> B i^i A == 0) -> (B i^i A == 0 <-> B C_ Compl A) -> (A i^i B == 0 <-> B C_ Compl A)
2		eqseq1	A i^i B == B i^i A -> (A i^i B == 0 <-> B i^i A == 0)
3		incom	A i^i B == B i^i A
4	2, 3	ax_mp	A i^i B == 0 <-> B i^i A == 0
5	1, 4	ax_mp	(B i^i A == 0 <-> B C_ Compl A) -> (A i^i B == 0 <-> B C_ Compl A)
6		ineq0	B i^i A == 0 <-> B C_ Compl A
7	5, 6	ax_mp	A i^i B == 0 <-> B C_ Compl 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)