Theorem abn0r 3439

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 2158	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	exbii 1598	. 2 |- ( E. x x e. { x | ph } <-> E. x ph )
3		nfab1 2314	. . 3 |- F/_ x { x | ph }
4	3	n0rf 3427	. 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 1485   e. wcel 2141   {cab 2156   =/= wne 2340   (/)c0 3414

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-nul 3415

This theorem is referenced by:  rabn0r  3441