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Theorem prtlem14 26742
Description: Lemma for prter1 26747, prter2 26749 and prtex 26748. (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 26740 . . 3  |-  ( Prt 
A  <->  A. x  e.  A  A. y  e.  A  ( x  =  y  \/  ( x  i^i  y
)  =  (/) ) )
2 rsp2 2605 . . 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 187 . 2  |-  ( Prt 
A  ->  ( (
x  e.  A  /\  y  e.  A )  ->  ( x  =  y  \/  ( x  i^i  y )  =  (/) ) ) )
4 elin 3358 . . . 4  |-  ( w  e.  ( x  i^i  y )  <->  ( w  e.  x  /\  w  e.  y ) )
5 eq0 3469 . . . . . 6  |-  ( ( x  i^i  y )  =  (/)  <->  A. w  -.  w  e.  ( x  i^i  y
) )
6 sp 1716 . . . . . 6  |-  ( A. w  -.  w  e.  ( x  i^i  y )  ->  -.  w  e.  ( x  i^i  y
) )
75, 6sylbi 187 . . . . 5  |-  ( ( x  i^i  y )  =  (/)  ->  -.  w  e.  ( x  i^i  y
) )
87pm2.21d 98 . . . 4  |-  ( ( x  i^i  y )  =  (/)  ->  ( w  e.  ( x  i^i  y )  ->  x  =  y ) )
94, 8syl5bir 209 . . 3  |-  ( ( x  i^i  y )  =  (/)  ->  ( ( w  e.  x  /\  w  e.  y )  ->  x  =  y ) )
109prtlem1 26707 . 2  |-  ( ( x  =  y  \/  ( x  i^i  y
)  =  (/) )  -> 
( ( w  e.  x  /\  w  e.  y )  ->  x  =  y ) )
113, 10syl6 29 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 357    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151   (/)c0 3455   Prt wprt 26739
This theorem is referenced by:  prtlem15  26743  prtlem17  26744
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-in 3159  df-nul 3456  df-prt 26740
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