Theorem abn0 2294

Description: Nonempty class abstraction.

Assertion

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

Proof of Theorem abn0

Step	Hyp	Ref	Expression
1		ne0 2292	. 2 |- ({x | ph} =/= (/) <-> E.y y e. {x | ph})
2		hbab1 1469	. . 3 |- (y e. {x | ph} -> A.x y e. {x | ph})
3		ax-17 973	. . 3 |- (x e. {x | ph} -> A.y x e. {x | ph})
4		eleq1 1537	. . 3 |- (y = x -> (y e. {x | ph} <-> x e. {x | ph}))
5	2, 3, 4	cbvex 1168	. 2 |- (E.y y e. {x | ph} <-> E.x x e. {x | ph})
6		abid 1468	. . 3 |- (x e. {x | ph} <-> ph)
7	6	exbii 1053	. 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 960  E.wex 982  {cab 1466   =/= wne 1588  (/)c0 2283

This theorem is referenced by:  rabn0 2296  intexab 2736  onminex 3026  relimasn 3431  fvprc 3727  fvopabn 3792  iinon 3916  oarec 4202  mapprc 4332  map0b 4349  map0 4350  pw2en 4452  scott0 4727  scott0s 4729  cp 4732  karden 4736  aceq3lem 4742  dffsum 6998  dfisum 7191  isumnul 7203  fine 10442

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-nul 2284

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