Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elex2VD Unicode version

Theorem elex2VD 28614
Description: Virtual deduction proof of elex2 2800. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elex2VD  |-  ( A  e.  B  ->  E. x  x  e.  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem elex2VD
StepHypRef Expression
1 idn1 28342 . . . . . 6  |-  (. A  e.  B  ->.  A  e.  B ).
2 idn2 28385 . . . . . 6  |-  (. A  e.  B ,. x  =  A  ->.  x  =  A ).
3 eleq1a 2352 . . . . . 6  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
41, 2, 3e12 28499 . . . . 5  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  B ).
54in2 28377 . . . 4  |-  (. A  e.  B  ->.  ( x  =  A  ->  x  e.  B ) ).
65gen11 28388 . . 3  |-  (. A  e.  B  ->.  A. x ( x  =  A  ->  x  e.  B ) ).
7 elisset 2798 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
81, 7e1_ 28399 . . 3  |-  (. A  e.  B  ->.  E. x  x  =  A ).
9 exim 1562 . . 3  |-  ( A. x ( x  =  A  ->  x  e.  B )  ->  ( E. x  x  =  A  ->  E. x  x  e.  B ) )
106, 8, 9e11 28460 . 2  |-  (. A  e.  B  ->.  E. x  x  e.  B ).
1110in1 28339 1  |-  ( A  e.  B  ->  E. x  x  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684
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-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790  df-vd1 28338  df-vd2 28347
  Copyright terms: Public domain W3C validator