Description: The empty set is not a member of a Cartesian product. (Contributed by NM, 2-May-1996) (Revised by Mario Carneiro, 26-Apr-2015) (Proof shortened by JJ, 13-Aug-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | 0nelxp | |- -. (/) e. ( A X. B ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex | |- x e. _V |
|
| 2 | vex | |- y e. _V |
|
| 3 | 1 2 | opnzi | |- <. x , y >. =/= (/) |
| 4 | 3 | nesymi | |- -. (/) = <. x , y >. |
| 5 | 4 | intnanr | |- -. ( (/) = <. x , y >. /\ ( x e. A /\ y e. B ) ) |
| 6 | 5 | nex | |- -. E. y ( (/) = <. x , y >. /\ ( x e. A /\ y e. B ) ) |
| 7 | 6 | nex | |- -. E. x E. y ( (/) = <. x , y >. /\ ( x e. A /\ y e. B ) ) |
| 8 | elxp | |- ( (/) e. ( A X. B ) <-> E. x E. y ( (/) = <. x , y >. /\ ( x e. A /\ y e. B ) ) ) |
|
| 9 | 7 8 | mtbir | |- -. (/) e. ( A X. B ) |