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Theorem prtlem400 32194
Description: Lemma for prter2 32205 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 2626 . 2  |-  -.  (/)  =/=  (/)
2 prtlem13.1 . . . 4  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
32prtlem16 32193 . . 3  |-  dom  .~  =  U. A
4 elqsn0 7431 . . 3  |-  ( ( dom  .~  =  U. A  /\  (/)  e.  ( U. A /.  .~  ) )  ->  (/)  =/=  (/) )
53, 4mpan 674 . 2  |-  ( (/)  e.  ( U. A /.  .~  )  ->  (/)  =/=  (/) )
61, 5mto 179 1  |-  -.  (/)  e.  ( U. A /.  .~  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   E.wrex 2774   (/)c0 3758   U.cuni 4213   {copab 4474   dom cdm 4845   /.cqs 7361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-xp 4851  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ec 7364  df-qs 7368
This theorem is referenced by:  prter2  32205
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