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Theorem rexn0 3556
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 3461 . . 3  |-  ( x  e.  A  ->  A  =/=  (/) )
21a1d 22 . 2  |-  ( x  e.  A  ->  ( ph  ->  A  =/=  (/) ) )
32rexlimiv 2661 1  |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    =/= wne 2446   E.wrex 2544   (/)c0 3455
This theorem is referenced by:  reusv2lem3  4537  reusv7OLD  4546  eusvobj2  6337  isdrs2  14073  ismnd  14369  slwn0  14926  lbsexg  15917  iuncon  17154  grpon0  20869  bsmgrli  25340  rngmgmbs3  25417  tpne  26075  filbcmb  26432  isbnd2  26507  rencldnfi  26904  stoweidlem14  27763  2reu4  27968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-nul 3456
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