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Theorem rexn0 3569
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 3474 . . 3  |-  ( x  e.  A  ->  A  =/=  (/) )
21a1d 22 . 2  |-  ( x  e.  A  ->  ( ph  ->  A  =/=  (/) ) )
32rexlimiv 2674 1  |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    =/= wne 2459   E.wrex 2557   (/)c0 3468
This theorem is referenced by:  reusv2lem3  4553  reusv7OLD  4562  eusvobj2  6353  isdrs2  14089  ismnd  14385  slwn0  14942  lbsexg  15933  iuncon  17170  grpon0  20885  bsmgrli  25443  rngmgmbs3  25520  tpne  26178  filbcmb  26535  isbnd2  26610  rencldnfi  27007  stoweidlem14  27866  2reu4  28071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-nul 3469
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