Theorem elxp ≪ | index | src | ≫

theorem elxp (A B: set) (a: nat): $ a e. Xp A B <-> fst a e. A /\ snd a e. B $;

Step	Hyp	Ref	Expression
1		bitr3	(fst a, snd a e. Xp A B <-> a e. Xp A B) -> (fst a, snd a e. Xp A B <-> fst a e. A /\ snd a e. B) -> (a e. Xp A B <-> fst a e. A /\ snd a e. B)
2		eleq1	fst a, snd a = a -> (fst a, snd a e. Xp A B <-> a e. Xp A B)
3		fstsnd	fst a, snd a = a
4	2, 3	ax_mp	fst a, snd a e. Xp A B <-> a e. Xp A B
5	1, 4	ax_mp	(fst a, snd a e. Xp A B <-> fst a e. A /\ snd a e. B) -> (a e. Xp A B <-> fst a e. A /\ snd a e. B)
6		prelxp	fst a, snd a e. Xp A B <-> fst a e. A /\ snd a e. B
7	5, 6	ax_mp	a e. Xp A B <-> fst a e. A /\ snd a e. B

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)