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Theorem rexn0 3673
Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
Assertion
Ref Expression
rexn0  |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem rexn0
StepHypRef Expression
1 ne0i 3577 . . 3  |-  ( x  e.  A  ->  A  =/=  (/) )
21a1d 23 . 2  |-  ( x  e.  A  ->  ( ph  ->  A  =/=  (/) ) )
32rexlimiv 2767 1  |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717    =/= wne 2550   E.wrex 2650   (/)c0 3571
This theorem is referenced by:  reusv2lem3  4666  reusv7OLD  4675  eusvobj2  6518  isdrs2  14323  ismnd  14619  slwn0  15176  lbsexg  16163  iuncon  17412  grpon0  21638  subofld  24071  filbcmb  26133  isbnd2  26183  rencldnfi  26573  stoweidlem14  27431  2reu4  27636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-v 2901  df-dif 3266  df-nul 3572
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