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Theorem opabn0 4311
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 3477 . 2  |-  ( {
<. x ,  y >.  |  ph }  =/=  (/)  <->  E. z 
z  e.  { <. x ,  y >.  |  ph } )
2 elopab 4288 . . . 4  |-  ( z  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) )
32exbii 1572 . . 3  |-  ( E. z  z  e.  { <. x ,  y >.  |  ph }  <->  E. z E. x E. y ( z  =  <. x ,  y >.  /\  ph ) )
4 exrot3 1830 . . . 4  |-  ( E. z E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y E. z ( z  =  <. x ,  y >.  /\  ph ) )
5 opex 4253 . . . . . . 7  |-  <. x ,  y >.  e.  _V
65isseti 2807 . . . . . 6  |-  E. z 
z  =  <. x ,  y >.
7 19.41v 1854 . . . . . 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 1573 . . . 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 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   <.cop 3656   {copab 4092
This theorem is referenced by:  dvdsrval  15443  thlle  16613  bcthlem5  18766  lgsquadlem3  20611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094
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