Description: The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017) Reduce axiom usage and shorten proof. (Revised by Gino Giotto, 3-Oct-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | 0nelopab | |- -. (/) e. { <. x , y >. | ph } |
| 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 >. /\ ph ) |
| 6 | 5 | nex | |- -. E. y ( (/) = <. x , y >. /\ ph ) |
| 7 | 6 | nex | |- -. E. x E. y ( (/) = <. x , y >. /\ ph ) |
| 8 | elopab | |- ( (/) e. { <. x , y >. | ph } <-> E. x E. y ( (/) = <. x , y >. /\ ph ) ) |
|
| 9 | 7 8 | mtbir | |- -. (/) e. { <. x , y >. | ph } |