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Theorem omnimkv 7147
Description: An omniscient set is Markov. In particular, the case where  A is  om means that the Limited Principle of Omniscience (LPO) implies Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)
Assertion
Ref Expression
omnimkv  |-  ( A  e. Omni  ->  A  e. Markov )

Proof of Theorem omnimkv
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isomni 7127 . . . 4  |-  ( A  e. Omni  ->  ( A  e. Omni  <->  A. f ( f : A --> 2o  ->  ( E. x  e.  A  ( f `  x
)  =  (/)  \/  A. x  e.  A  (
f `  x )  =  1o ) ) ) )
21ibi 176 . . 3  |-  ( A  e. Omni  ->  A. f ( f : A --> 2o  ->  ( E. x  e.  A  ( f `  x
)  =  (/)  \/  A. x  e.  A  (
f `  x )  =  1o ) ) )
3 pm2.53 722 . . . . . . 7  |-  ( ( A. x  e.  A  ( f `  x
)  =  1o  \/  E. x  e.  A  ( f `  x )  =  (/) )  ->  ( -.  A. x  e.  A  ( f `  x
)  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) )
43orcoms 730 . . . . . 6  |-  ( ( E. x  e.  A  ( f `  x
)  =  (/)  \/  A. x  e.  A  (
f `  x )  =  1o )  ->  ( -.  A. x  e.  A  ( f `  x
)  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) )
54a1i 9 . . . . 5  |-  ( A  e. Omni  ->  ( ( E. x  e.  A  ( f `  x )  =  (/)  \/  A. x  e.  A  ( f `  x )  =  1o )  ->  ( -.  A. x  e.  A  ( f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) ) )
65imim2d 54 . . . 4  |-  ( A  e. Omni  ->  ( ( f : A --> 2o  ->  ( E. x  e.  A  ( f `  x
)  =  (/)  \/  A. x  e.  A  (
f `  x )  =  1o ) )  -> 
( f : A --> 2o  ->  ( -.  A. x  e.  A  (
f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) ) ) )
76alimdv 1879 . . 3  |-  ( A  e. Omni  ->  ( A. f
( f : A --> 2o  ->  ( E. x  e.  A  ( f `  x )  =  (/)  \/ 
A. x  e.  A  ( f `  x
)  =  1o ) )  ->  A. f
( f : A --> 2o  ->  ( -.  A. x  e.  A  (
f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) ) ) )
82, 7mpd 13 . 2  |-  ( A  e. Omni  ->  A. f ( f : A --> 2o  ->  ( -.  A. x  e.  A  ( f `  x )  =  1o 
->  E. x  e.  A  ( f `  x
)  =  (/) ) ) )
9 ismkv 7144 . 2  |-  ( A  e. Omni  ->  ( A  e. Markov  <->  A. f ( f : A --> 2o  ->  ( -.  A. x  e.  A  ( f `  x
)  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) ) ) )
108, 9mpbird 167 1  |-  ( A  e. Omni  ->  A  e. Markov )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 708   A.wal 1351    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   (/)c0 3422   -->wf 5207   ` cfv 5211   1oc1o 6403   2oc2o 6404  Omnicomni 7125  Markovcmarkov 7142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-fn 5214  df-f 5215  df-omni 7126  df-markov 7143
This theorem is referenced by:  exmidmp  7148
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