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Theorem rabn0r 3487
Description: Nonempty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
Assertion
Ref Expression
rabn0r  |-  ( E. x  e.  A  ph  ->  { x  e.  A  |  ph }  =/=  (/) )

Proof of Theorem rabn0r
StepHypRef Expression
1 abn0r 3485 . 2  |-  ( E. x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  =/=  (/) )
2 df-rex 2490 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
3 df-rab 2493 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43neeq1i 2391 . 2  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  { x  |  ( x  e.  A  /\  ph ) }  =/=  (/) )
51, 2, 43imtr4i 201 1  |-  ( E. x  e.  A  ph  ->  { x  e.  A  |  ph }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   E.wex 1515    e. wcel 2176   {cab 2191    =/= wne 2376   E.wrex 2485   {crab 2488   (/)c0 3460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-nul 3461
This theorem is referenced by:  sgmnncl  15460
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