Theorem xpss1 ≪ | index | src | ≫

theorem xpss1 (A B C: set): $ A C_ B -> Xp A C C_ Xp B C $;

Step	Hyp	Ref	Expression
1		elxp	p e. Xp A C <-> fst p e. A /\ snd p e. C
2		elxp	p e. Xp B C <-> fst p e. B /\ snd p e. C
3	1, 2	imeqi	p e. Xp A C -> p e. Xp B C <-> fst p e. A /\ snd p e. C -> fst p e. B /\ snd p e. C
4		ssel	A C_ B -> fst p e. A -> fst p e. B
5	4	anim1d	A C_ B -> fst p e. A /\ snd p e. C -> fst p e. B /\ snd p e. C
6	3, 5	sylibr	A C_ B -> p e. Xp A C -> p e. Xp B C
7	6	iald	A C_ B -> A. p (p e. Xp A C -> p e. Xp B C)
8	7	conv subset	A C_ B -> Xp A C C_ Xp B 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)