Theorem abn0r 3392

Description: Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)

Assertion

Ref	Expression
abn0r	|- ( E. x ph -> { x | ph } =/= (/) )

Proof of Theorem abn0r

Step	Hyp	Ref	Expression
1		abid 2128	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	exbii 1585	. 2 |- ( E. x x e. { x | ph } <-> E. x ph )
3		nfab1 2284	. . 3 |- F/_ x { x | ph }
4	3	n0rf 3380	. 2 |- ( E. x x e. { x | ph } -> { x | ph } =/= (/) )
5	2, 4	sylbir 134	1 |- ( E. x ph -> { x | ph } =/= (/) )

Syntax hints:   -> wi 4   E.wex 1469   e. wcel 1481   {cab 2126   =/= wne 2309   (/)c0 3368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-dif 3078  df-nul 3369

This theorem is referenced by:  rabn0r  3394