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Theorem elrabi 2718
Description: Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
Assertion
Ref Expression
elrabi  |-  ( A  e.  { x  e.  V  |  ph }  ->  A  e.  V )
Distinct variable groups:    x, A    x, V
Allowed substitution hint:    ph( x)

Proof of Theorem elrabi
StepHypRef Expression
1 clelab 2178 . . 3  |-  ( A  e.  { x  |  ( x  e.  V  /\  ph ) }  <->  E. x
( x  =  A  /\  ( x  e.  V  /\  ph )
) )
2 eleq1 2116 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  V  <->  A  e.  V ) )
32anbi1d 446 . . . . 5  |-  ( x  =  A  ->  (
( x  e.  V  /\  ph )  <->  ( A  e.  V  /\  ph )
) )
43simprbda 369 . . . 4  |-  ( ( x  =  A  /\  ( x  e.  V  /\  ph ) )  ->  A  e.  V )
54exlimiv 1505 . . 3  |-  ( E. x ( x  =  A  /\  ( x  e.  V  /\  ph ) )  ->  A  e.  V )
61, 5sylbi 118 . 2  |-  ( A  e.  { x  |  ( x  e.  V  /\  ph ) }  ->  A  e.  V )
7 df-rab 2332 . 2  |-  { x  e.  V  |  ph }  =  { x  |  ( x  e.  V  /\  ph ) }
86, 7eleq2s 2148 1  |-  ( A  e.  { x  e.  V  |  ph }  ->  A  e.  V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042   {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-rab 2332
This theorem is referenced by:  ordtriexmidlem  4273  ordtri2or2exmidlem  4279  onsucelsucexmidlem  4282  ordsoexmid  4314  reg3exmidlemwe  4331  acexmidlemcase  5535  genpelvl  6668  genpelvu  6669  nnindnn  7025  nnind  8006  ublbneg  8645
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