Description: A positive real is not empty. (Contributed by NM, 15-May-1996) (Revised by Mario Carneiro, 11-May-2013) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | prn0 | |- ( A e. P. -> A =/= (/) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnpi | |- ( A e. P. <-> ( ( A e. _V /\ (/) C. A /\ A C. Q. ) /\ A. x e. A ( A. y ( yy e. A ) /\ E. y e. A x |
|
2 | simpl2 | |- ( ( ( A e. _V /\ (/) C. A /\ A C. Q. ) /\ A. x e. A ( A. y ( yy e. A ) /\ E. y e. A x(/) C. A ) |
|
3 | 1 2 | sylbi | |- ( A e. P. -> (/) C. A ) |
4 | 0pss | |- ( (/) C. A <-> A =/= (/) ) |
|
5 | 3 4 | sylib | |- ( A e. P. -> A =/= (/) ) |