Theorem abn0r 3387

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 2127	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	exbii 1584	. 2 |- ( E. x x e. { x | ph } <-> E. x ph )
3		nfab1 2283	. . 3 |- F/_ x { x | ph }
4	3	n0rf 3375	. 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 1468   e. wcel 1480   {cab 2125   =/= wne 2308   (/)c0 3363

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-dif 3073  df-nul 3364

This theorem is referenced by:  rabn0r  3389