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Theorem prtlem400 26720
Description: Lemma for prter2 26731 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 2607 . 2  |-  -.  (/)  =/=  (/)
2 prtlem13.1 . . . 4  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
32prtlem16 26719 . . 3  |-  dom  .~  =  U. A
4 elqsn0 6974 . . 3  |-  ( ( dom  .~  =  U. A  /\  (/)  e.  ( U. A /.  .~  ) )  ->  (/)  =/=  (/) )
53, 4mpan 653 . 2  |-  ( (/)  e.  ( U. A /.  .~  )  ->  (/)  =/=  (/) )
61, 5mto 170 1  |-  -.  (/)  e.  ( U. A /.  .~  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707   (/)c0 3629   U.cuni 4016   {copab 4266   dom cdm 4879   /.cqs 6905
This theorem is referenced by:  prter2  26731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-xp 4885  df-cnv 4887  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-ec 6908  df-qs 6912
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