Theorem disjne ≪ | index | src | ≫

theorem disjne (A B: set) (G: wff) (x y: nat):
  $ G -> A i^i B == 0 $ >
  $ G -> x e. A $ >
  $ G -> y e. B $ >
  $ G -> x != y $;

Step	Hyp	Ref	Expression
1		el02	~y e. 0
2		hyp h1	G -> A i^i B == 0
3	2	anwl	G /\ x = y -> A i^i B == 0
4	3	eleq2d	G /\ x = y -> (y e. A i^i B <-> y e. 0)
5		elin	y e. A i^i B <-> y e. A /\ y e. B
6		anr	G /\ x = y -> x = y
7	6	eleq1d	G /\ x = y -> (x e. A <-> y e. A)
8		hyp h2	G -> x e. A
9	8	anwl	G /\ x = y -> x e. A
10	7, 9	mpbid	G /\ x = y -> y e. A
11		hyp h3	G -> y e. B
12	11	anwl	G /\ x = y -> y e. B
13	10, 12	iand	G /\ x = y -> y e. A /\ y e. B
14	5, 13	sylibr	G /\ x = y -> y e. A i^i B
15	4, 14	mpbid	G /\ x = y -> y e. 0
16	15	exp	G -> x = y -> y e. 0
17	16	con3d	G -> ~y e. 0 -> ~x = y
18	17	conv ne	G -> ~y e. 0 -> x != y
19	1, 18	mpi	G -> x != y

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)