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Theorem exmidac 7205
Description: The axiom of choice implies excluded middle. See acexmid 5871 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.)
Assertion
Ref Expression
exmidac  |-  (CHOICE  -> EXMID )

Proof of Theorem exmidac
Dummy variables  x  u  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2184 . . . 4  |-  ( u  =  x  ->  (
u  =  (/)  <->  x  =  (/) ) )
21orbi1d 791 . . 3  |-  ( u  =  x  ->  (
( u  =  (/)  \/  y  =  { (/) } )  <->  ( x  =  (/)  \/  y  =  { (/)
} ) ) )
32cbvrabv 2736 . 2  |-  { u  e.  { (/) ,  { (/) } }  |  ( u  =  (/)  \/  y  =  { (/) } ) }  =  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  y  =  { (/) } ) }
4 eqeq1 2184 . . . 4  |-  ( u  =  x  ->  (
u  =  { (/) }  <-> 
x  =  { (/) } ) )
54orbi1d 791 . . 3  |-  ( u  =  x  ->  (
( u  =  { (/)
}  \/  y  =  { (/) } )  <->  ( x  =  { (/) }  \/  y  =  { (/) } ) ) )
65cbvrabv 2736 . 2  |-  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  y  =  { (/) } ) }  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/) }  \/  y  =  { (/) } ) }
7 eqid 2177 . 2  |-  { {
u  e.  { (/) ,  { (/) } }  | 
( u  =  (/)  \/  y  =  { (/) } ) } ,  {
u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  y  =  { (/) } ) } }  =  { {
u  e.  { (/) ,  { (/) } }  | 
( u  =  (/)  \/  y  =  { (/) } ) } ,  {
u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  y  =  { (/) } ) } }
83, 6, 7exmidaclem 7204 1  |-  (CHOICE  -> EXMID )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 708    = wceq 1353   {crab 2459   (/)c0 3422   {csn 3592   {cpr 3593  EXMIDwem 4193  CHOICEwac 7201
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-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-exmid 4194  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-ac 7202
This theorem is referenced by: (None)
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