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Theorem opabn0 4705
Description: Nonempty 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 3709 . 2  |-  ( {
<. x ,  y >.  |  ph }  =/=  (/)  <->  E. z 
z  e.  { <. x ,  y >.  |  ph } )
2 elopab 4682 . . . 4  |-  ( z  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) )
32exbii 1722 . . 3  |-  ( E. z  z  e.  { <. x ,  y >.  |  ph }  <->  E. z E. x E. y ( z  =  <. x ,  y >.  /\  ph ) )
4 exrot3 1935 . . . 4  |-  ( E. z E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y E. z ( z  =  <. x ,  y >.  /\  ph ) )
5 opex 4637 . . . . . . 7  |-  <. x ,  y >.  e.  _V
65isseti 3019 . . . . . 6  |-  E. z 
z  =  <. x ,  y >.
7 19.41v 1834 . . . . . 6  |-  ( E. z ( z  = 
<. x ,  y >.  /\  ph )  <->  ( E. z  z  =  <. x ,  y >.  /\  ph ) )
86, 7mpbiran 929 . . . . 5  |-  ( E. z ( z  = 
<. x ,  y >.  /\  ph )  <->  ph )
982exbii 1723 . . . 4  |-  ( E. x E. y E. z ( z  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
104, 9bitri 257 . . 3  |-  ( E. z E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
113, 10bitri 257 . 2  |-  ( E. z  z  e.  { <. x ,  y >.  |  ph }  <->  E. x E. y ph )
121, 11bitri 257 1  |-  ( {
<. x ,  y >.  |  ph }  =/=  (/)  <->  E. x E. y ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    = wceq 1448   E.wex 1667    e. wcel 1891    =/= wne 2622   (/)c0 3699   <.cop 3942   {copab 4432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pr 4612
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-v 3015  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-opab 4434
This theorem is referenced by:  csbopab  4706  dvdsrval  17884  thlle  19271  bcthlem5  22307  lgsquadlem3  24296
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