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Theorem ballotlemoex 23044
Description:  O is a set. (Contributed by Thierry Arnoux, 7-Dec-2016.)
Hypotheses
Ref Expression
ballotth.m  |-  M  e.  NN
ballotth.n  |-  N  e.  NN
ballotth.o  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
Assertion
Ref Expression
ballotlemoex  |-  O  e. 
_V
Distinct variable groups:    M, c    N, c    O, c

Proof of Theorem ballotlemoex
StepHypRef Expression
1 ovex 5883 . . 3  |-  ( 1 ... ( M  +  N ) )  e. 
_V
21pwex 4193 . 2  |-  ~P (
1 ... ( M  +  N ) )  e. 
_V
3 ballotth.o . . 3  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 ssrab2 3258 . . 3  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  C_ 
~P ( 1 ... ( M  +  N
) )
53, 4eqsstri 3208 . 2  |-  O  C_  ~P ( 1 ... ( M  +  N )
)
62, 5ssexi 4159 1  |-  O  e. 
_V
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   ~Pcpw 3625   ` cfv 5255  (class class class)co 5858   1c1 8738    + caddc 8740   NNcn 9746   ...cfz 10782   #chash 11337
This theorem is referenced by:  ballotlem2  23047  ballotlem8  23095
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188
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-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219  df-fv 5263  df-ov 5861
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