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Theorem prtlem400 30442
Description: Lemma for prter2 30453 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem13.1  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
Assertion
Ref Expression
prtlem400  |-  -.  (/)  e.  ( U. A /.  .~  )
Distinct variable group:    x, u, y, A
Allowed substitution hints:    .~ ( x, y, u)

Proof of Theorem prtlem400
StepHypRef Expression
1 neirr 2671 . 2  |-  -.  (/)  =/=  (/)
2 prtlem13.1 . . . 4  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
32prtlem16 30441 . . 3  |-  dom  .~  =  U. A
4 elqsn0 7381 . . 3  |-  ( ( dom  .~  =  U. A  /\  (/)  e.  ( U. A /.  .~  ) )  ->  (/)  =/=  (/) )
53, 4mpan 670 . 2  |-  ( (/)  e.  ( U. A /.  .~  )  ->  (/)  =/=  (/) )
61, 5mto 176 1  |-  -.  (/)  e.  ( U. A /.  .~  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   (/)c0 3785   U.cuni 4245   {copab 4504   dom cdm 4999   /.cqs 7311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-ec 7314  df-qs 7318
This theorem is referenced by:  prter2  30453
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