Theorem intexab 2731

Description: The intersection of a non-empty class abstraction exists.

Assertion

Ref	Expression
intexab	|- (E.xph <-> |^|{x | ph} e. V)

Proof of Theorem intexab

Step	Hyp	Ref	Expression
1		abn0 2290	. 2 |- ({x | ph} =/= (/) <-> E.xph)
2		intex 2729	. 2 |- ({x | ph} =/= (/) <-> |^|{x | ph} e. V)
3	1, 2	bitr3 175	1 |- (E.xph <-> |^|{x | ph} e. V)

Syntax hints:   <-> wb 146   e. wcel 958  E.wex 980  {cab 1463   =/= wne 1585  Vcvv 1811  (/)c0 2280  |^|cint 2533

This theorem is referenced by:  intexrab 2732  cfval 4906  cffnon 4907

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281  df-int 2534

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