Theorem abn0r 3243

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 2028	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	exbii 1496	. 2 |- ( E. x x e. { x | ph } <-> E. x ph )
3		nfab1 2180	. . 3 |- F/_ x { x | ph }
4	3	n0rf 3233	. 2 |- ( E. x x e. { x | ph } -> { x | ph } =/= (/) )
5	2, 4	sylbir 125	1 |- ( E. x ph -> { x | ph } =/= (/) )

Syntax hints:   -> wi 4   E.wex 1381   e. wcel 1393   {cab 2026   =/= wne 2204   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-nul 3225

This theorem is referenced by:  rabn0r  3244