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Theorem prtlem18 26726
Description: Lemma for prter2 26730. (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 2767 . . . . 5  |-  ( ( v  e.  A  /\  ( z  e.  v  /\  w  e.  v ) )  ->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
21expr 599 . . . 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 26717 . . . 4  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
52, 4syl6ibr 219 . . 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 26717 . . 3  |-  ( z  .~  w  <->  E. p  e.  A  ( z  e.  p  /\  w  e.  p ) )
8 prtlem17 26725 . . 3  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( E. p  e.  A  ( z  e.  p  /\  w  e.  p )  ->  w  e.  v ) ) )
97, 8syl7bi 222 . 2  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( z  .~  w  ->  w  e.  v ) ) )
106, 9impbidd 182 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   class class class wbr 4212   {copab 4265   Prt wprt 26720
This theorem is referenced by:  prtlem19  26727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-prt 26721
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