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Theorem opabn0 4295
Description: Non-empty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.)
Assertion
Ref Expression
opabn0  |-  ( {
<. x ,  y >.  |  ph }  =/=  (/)  <->  E. x E. y ph )

Proof of Theorem opabn0
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 n0 3464 . 2  |-  ( {
<. x ,  y >.  |  ph }  =/=  (/)  <->  E. z 
z  e.  { <. x ,  y >.  |  ph } )
2 elopab 4272 . . . 4  |-  ( z  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) )
32exbii 1569 . . 3  |-  ( E. z  z  e.  { <. x ,  y >.  |  ph }  <->  E. z E. x E. y ( z  =  <. x ,  y >.  /\  ph ) )
4 exrot3 1818 . . . 4  |-  ( E. z E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y E. z ( z  =  <. x ,  y >.  /\  ph ) )
5 opex 4237 . . . . . . 7  |-  <. x ,  y >.  e.  _V
65isseti 2794 . . . . . 6  |-  E. z 
z  =  <. x ,  y >.
7 19.41v 1842 . . . . . 6  |-  ( E. z ( z  = 
<. x ,  y >.  /\  ph )  <->  ( E. z  z  =  <. x ,  y >.  /\  ph ) )
86, 7mpbiran 884 . . . . 5  |-  ( E. z ( z  = 
<. x ,  y >.  /\  ph )  <->  ph )
982exbii 1570 . . . 4  |-  ( E. x E. y E. z ( z  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
104, 9bitri 240 . . 3  |-  ( E. z E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
113, 10bitri 240 . 2  |-  ( E. z  z  e.  { <. x ,  y >.  |  ph }  <->  E. x E. y ph )
121, 11bitri 240 1  |-  ( {
<. x ,  y >.  |  ph }  =/=  (/)  <->  E. x E. y ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   <.cop 3643   {copab 4076
This theorem is referenced by:  dvdsrval  15427  thlle  16597  bcthlem5  18750  lgsquadlem3  20595
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078
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