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Theorem prtlem18 28947
Description: Lemma for prter2 28951. (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 2775 . . . . 5  |-  ( ( v  e.  A  /\  ( z  e.  v  /\  w  e.  v ) )  ->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
21expr 612 . . . 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 28938 . . . 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 28938 . . 3  |-  ( z  .~  w  <->  E. p  e.  A  ( z  e.  p  /\  w  e.  p ) )
8 prtlem17 28946 . . 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 1364    e. wcel 1761   E.wrex 2714   class class class wbr 4289   {copab 4346   Prt wprt 28941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-prt 28942
This theorem is referenced by:  prtlem19  28948
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