Theorem intexab 4822

Description: The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.)

Assertion

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

Proof of Theorem intexab

Step	Hyp	Ref	Expression
1		abn0 3954	. 2 |- ( { x | ph } =/= (/) <-> E. x ph )
2		intex 4820	. 2 |- ( { x | ph } =/= (/) <-> |^| { x | ph } e. _V )
3	1, 2	bitr3i 266	1 |- ( E. x ph <-> |^| { x | ph } e. _V )

Syntax hints:   <-> wb 196   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   _Vcvv 3200   (/)c0 3915   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-int 4476

This theorem is referenced by:  intexrab  4823  tcmin  8617  cfval  9069  efgval  18130  relintabex  37887  rclexi  37922  rtrclex  37924  trclexi  37927  rtrclexi  37928