Theorem xpss2 ≪ | index | src | ≫

theorem xpss2 (A B C: set): $ B C_ C -> Xp A B C_ Xp A C $;

Step	Hyp	Ref	Expression
1		elxp	p e. Xp A B <-> fst p e. A /\ snd p e. B
2		elxp	p e. Xp A C <-> fst p e. A /\ snd p e. C
3	1, 2	imeqi	p e. Xp A B -> p e. Xp A C <-> fst p e. A /\ snd p e. B -> fst p e. A /\ snd p e. C
4		ssel	B C_ C -> snd p e. B -> snd p e. C
5	4	anim2d	B C_ C -> fst p e. A /\ snd p e. B -> fst p e. A /\ snd p e. C
6	3, 5	sylibr	B C_ C -> p e. Xp A B -> p e. Xp A C
7	6	iald	B C_ C -> A. p (p e. Xp A B -> p e. Xp A C)
8	7	conv subset	B C_ C -> Xp A B C_ Xp A C

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)