Theorem abn0 2280

Description: Nonempty class abstraction.

Assertion

Ref	Expression
abn0	|- ({x | ph} =/= (/) <-> E.xph)

Proof of Theorem abn0

Step	Hyp	Ref	Expression
1		ne0 2278	. 2 |- ({x | ph} =/= (/) <-> E.y y e. {x | ph})
2		hbab1 1459	. . 3 |- (y e. {x | ph} -> A.x y e. {x | ph})
3		ax-17 968	. . 3 |- (x e. {x | ph} -> A.y x e. {x | ph})
4		eleq1 1526	. . 3 |- (y = x -> (y e. {x | ph} <-> x e. {x | ph}))
5	2, 3, 4	cbvex 1162	. 2 |- (E.y y e. {x | ph} <-> E.x x e. {x | ph})
6		abid 1458	. . 3 |- (x e. {x | ph} <-> ph)
7	6	exbii 1047	. 2 |- (E.x x e. {x | ph} <-> E.xph)
8	1, 5, 7	3bitr 177	1 |- ({x | ph} =/= (/) <-> E.xph)

Syntax hints:   <-> wb 146   e. wcel 955  E.wex 977  {cab 1456   =/= wne 1577  (/)c0 2270

This theorem is referenced by:  rabn0 2282  intexab 2721  onminex 3010  relimasn 3409  fvprc 3706  fvopabn 3771  iinon 3895  oarec 4180  mapprc 4310  map0b 4327  map0 4328  pw2en 4426  scott0 4689  scott0s 4691  cp 4694  karden 4698  aceq3lem 4704  dffsum 6936  dfisum 7127  isumnul 7138  fine 10348

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-nul 2271

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