Theorem inincpl ≪ | index | src | ≫

theorem inincpl (A B: set): $ A i^i (B i^i Compl A) == 0 $;

Step	Hyp	Ref	Expression
1		sseq0	A i^i (B i^i Compl A) C_ A i^i Compl A -> A i^i Compl A == 0 -> A i^i (B i^i Compl A) == 0
2		ssin2	B i^i Compl A C_ Compl A -> A i^i (B i^i Compl A) C_ A i^i Compl A
3		inss2	B i^i Compl A C_ Compl A
4	2, 3	ax_mp	A i^i (B i^i Compl A) C_ A i^i Compl A
5	1, 4	ax_mp	A i^i Compl A == 0 -> A i^i (B i^i Compl A) == 0
6		incpl2	A i^i Compl A == 0
7	5, 6	ax_mp	A i^i (B i^i Compl A) == 0

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)