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Theorem opabn0 4488
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 3639 . 2  |-  ( {
<. x ,  y >.  |  ph }  =/=  (/)  <->  E. z 
z  e.  { <. x ,  y >.  |  ph } )
2 elopab 4465 . . . 4  |-  ( z  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) )
32exbii 1593 . . 3  |-  ( E. z  z  e.  { <. x ,  y >.  |  ph }  <->  E. z E. x E. y ( z  =  <. x ,  y >.  /\  ph ) )
4 exrot3 1760 . . . 4  |-  ( E. z E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y E. z ( z  =  <. x ,  y >.  /\  ph ) )
5 opex 4430 . . . . . . 7  |-  <. x ,  y >.  e.  _V
65isseti 2964 . . . . . 6  |-  E. z 
z  =  <. x ,  y >.
7 19.41v 1925 . . . . . 6  |-  ( E. z ( z  = 
<. x ,  y >.  /\  ph )  <->  ( E. z  z  =  <. x ,  y >.  /\  ph ) )
86, 7mpbiran 886 . . . . 5  |-  ( E. z ( z  = 
<. x ,  y >.  /\  ph )  <->  ph )
982exbii 1594 . . . 4  |-  ( E. x E. y E. z ( z  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
104, 9bitri 242 . . 3  |-  ( E. z E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
113, 10bitri 242 . 2  |-  ( E. z  z  e.  { <. x ,  y >.  |  ph }  <->  E. x E. y ph )
121, 11bitri 242 1  |-  ( {
<. x ,  y >.  |  ph }  =/=  (/)  <->  E. x E. y ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   (/)c0 3630   <.cop 3819   {copab 4268
This theorem is referenced by:  dvdsrval  15755  thlle  16929  bcthlem5  19286  lgsquadlem3  21145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4270
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