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Theorem prtlem18 25893
Description: Lemma for prter2 25897. (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 2638 . . . . 5  |-  ( ( v  e.  A  /\  ( z  e.  v  /\  w  e.  v ) )  ->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
21expr 598 . . . 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 25884 . . . 4  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
52, 4syl6ibr 218 . . 3  |-  ( ( v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  ->  z  .~  w
) )
65a1i 10 . 2  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  ->  z  .~  w
) ) )
73prtlem13 25884 . . 3  |-  ( z  .~  w  <->  E. p  e.  A  ( z  e.  p  /\  w  e.  p ) )
8 prtlem17 25892 . . 3  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( E. p  e.  A  ( z  e.  p  /\  w  e.  p )  ->  w  e.  v ) ) )
97, 8syl7bi 221 . 2  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( z  .~  w  ->  w  e.  v ) ) )
106, 9impbidd 181 1  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  <-> 
z  .~  w )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   E.wrex 2578   class class class wbr 4060   {copab 4113   Prt wprt 25887
This theorem is referenced by:  prtlem19  25894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-prt 25888
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