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Theorem prtlem14 26723
Description: Lemma for prter1 26728, prter2 26730 and prtex 26729. (Contributed by Rodolfo Medina, 13-Oct-2010.)
Assertion
Ref Expression
prtlem14  |-  ( Prt 
A  ->  ( (
x  e.  A  /\  y  e.  A )  ->  ( ( w  e.  x  /\  w  e.  y )  ->  x  =  y ) ) )
Distinct variable groups:    x, w, y    x, A, y
Allowed substitution hint:    A( w)

Proof of Theorem prtlem14
StepHypRef Expression
1 df-prt 26721 . . 3  |-  ( Prt 
A  <->  A. x  e.  A  A. y  e.  A  ( x  =  y  \/  ( x  i^i  y
)  =  (/) ) )
2 rsp2 2768 . . 3  |-  ( A. x  e.  A  A. y  e.  A  (
x  =  y  \/  ( x  i^i  y
)  =  (/) )  -> 
( ( x  e.  A  /\  y  e.  A )  ->  (
x  =  y  \/  ( x  i^i  y
)  =  (/) ) ) )
31, 2sylbi 188 . 2  |-  ( Prt 
A  ->  ( (
x  e.  A  /\  y  e.  A )  ->  ( x  =  y  \/  ( x  i^i  y )  =  (/) ) ) )
4 elin 3530 . . . 4  |-  ( w  e.  ( x  i^i  y )  <->  ( w  e.  x  /\  w  e.  y ) )
5 eq0 3642 . . . . . 6  |-  ( ( x  i^i  y )  =  (/)  <->  A. w  -.  w  e.  ( x  i^i  y
) )
6 sp 1763 . . . . . 6  |-  ( A. w  -.  w  e.  ( x  i^i  y )  ->  -.  w  e.  ( x  i^i  y
) )
75, 6sylbi 188 . . . . 5  |-  ( ( x  i^i  y )  =  (/)  ->  -.  w  e.  ( x  i^i  y
) )
87pm2.21d 100 . . . 4  |-  ( ( x  i^i  y )  =  (/)  ->  ( w  e.  ( x  i^i  y )  ->  x  =  y ) )
94, 8syl5bir 210 . . 3  |-  ( ( x  i^i  y )  =  (/)  ->  ( ( w  e.  x  /\  w  e.  y )  ->  x  =  y ) )
109prtlem1 26690 . 2  |-  ( ( x  =  y  \/  ( x  i^i  y
)  =  (/) )  -> 
( ( w  e.  x  /\  w  e.  y )  ->  x  =  y ) )
113, 10syl6 31 1  |-  ( Prt 
A  ->  ( (
x  e.  A  /\  y  e.  A )  ->  ( ( w  e.  x  /\  w  e.  y )  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   A.wral 2705    i^i cin 3319   (/)c0 3628   Prt wprt 26720
This theorem is referenced by:  prtlem15  26724  prtlem17  26725
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-v 2958  df-dif 3323  df-in 3327  df-nul 3629  df-prt 26721
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