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Theorem prtlem18 30250
Description: Lemma for prter2 30254. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem18.1  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
Assertion
Ref Expression
prtlem18  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  <-> 
z  .~  w )
) )
Distinct variable groups:    v, u, w, x, y, z, A   
v,  .~ , w, z
Allowed substitution hints:    .~ ( x, y, u)

Proof of Theorem prtlem18
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 rspe 2922 . . . . 5  |-  ( ( v  e.  A  /\  ( z  e.  v  /\  w  e.  v ) )  ->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
21expr 615 . . . 4  |-  ( ( v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  ->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) ) )
3 prtlem18.1 . . . . 5  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
43prtlem13 30241 . . . 4  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
52, 4syl6ibr 227 . . 3  |-  ( ( v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  ->  z  .~  w
) )
65a1i 11 . 2  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  ->  z  .~  w
) ) )
73prtlem13 30241 . . 3  |-  ( z  .~  w  <->  E. p  e.  A  ( z  e.  p  /\  w  e.  p ) )
8 prtlem17 30249 . . 3  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( E. p  e.  A  ( z  e.  p  /\  w  e.  p )  ->  w  e.  v ) ) )
97, 8syl7bi 230 . 2  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( z  .~  w  ->  w  e.  v ) ) )
106, 9impbidd 189 1  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  <-> 
z  .~  w )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   class class class wbr 4447   {copab 4504   Prt wprt 30244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-prt 30245
This theorem is referenced by:  prtlem19  30251
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