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Theorem ballotlemoex 24748
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 6109 . . 3  |-  ( 1 ... ( M  +  N ) )  e. 
_V
21pwex 4385 . 2  |-  ~P (
1 ... ( M  +  N ) )  e. 
_V
3 ballotth.o . . 3  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 ssrab2 3430 . . 3  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  C_ 
~P ( 1 ... ( M  +  N
) )
53, 4eqsstri 3380 . 2  |-  O  C_  ~P ( 1 ... ( M  +  N )
)
62, 5ssexi 4351 1  |-  O  e. 
_V
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958   ~Pcpw 3801   ` cfv 5457  (class class class)co 6084   1c1 8996    + caddc 8998   NNcn 10005   ...cfz 11048   #chash 11623
This theorem is referenced by:  ballotlem2  24751  ballotlem8  24799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-pw 3803  df-sn 3822  df-pr 3823  df-uni 4018  df-iota 5421  df-fv 5465  df-ov 6087
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