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Theorem prtlem400 26841
Description: Lemma for prter2 26852 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 2464 . 2  |-  -.  (/)  =/=  (/)
2 prtlem13.1 . . . 4  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
32prtlem16 26840 . . 3  |-  dom  .~  =  U. A
4 elqsn0 6744 . . 3  |-  ( ( dom  .~  =  U. A  /\  (/)  e.  ( U. A /.  .~  ) )  ->  (/)  =/=  (/) )
53, 4mpan 651 . 2  |-  ( (/)  e.  ( U. A /.  .~  )  ->  (/)  =/=  (/) )
61, 5mto 167 1  |-  -.  (/)  e.  ( U. A /.  .~  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   (/)c0 3468   U.cuni 3843   {copab 4092   dom cdm 4705   /.cqs 6675
This theorem is referenced by:  prter2  26852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-ec 6678  df-qs 6682
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