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Theorem ballotlemoex 23917
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 5925 . . 3  |-  ( 1 ... ( M  +  N ) )  e. 
_V
21pwex 4230 . 2  |-  ~P (
1 ... ( M  +  N ) )  e. 
_V
3 ballotth.o . . 3  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 ssrab2 3292 . . 3  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  C_ 
~P ( 1 ... ( M  +  N
) )
53, 4eqsstri 3242 . 2  |-  O  C_  ~P ( 1 ... ( M  +  N )
)
62, 5ssexi 4196 1  |-  O  e. 
_V
Colors of variables: wff set class
Syntax hints:    = wceq 1633    e. wcel 1701   {crab 2581   _Vcvv 2822   ~Pcpw 3659   ` cfv 5292  (class class class)co 5900   1c1 8783    + caddc 8785   NNcn 9791   ...cfz 10829   #chash 11384
This theorem is referenced by:  ballotlem2  23920  ballotlem8  23968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-pw 3661  df-sn 3680  df-pr 3681  df-uni 3865  df-iota 5256  df-fv 5300  df-ov 5903
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