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Theorem exmidpweq 6911
Description: Excluded middle is equivalent to the power set of  1o being  2o. (Contributed by Jim Kingdon, 28-Jul-2024.)
Assertion
Ref Expression
exmidpweq  |-  (EXMID  <->  ~P 1o  =  2o )

Proof of Theorem exmidpweq
StepHypRef Expression
1 exmid01 4200 . . . . . . . 8  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
21biimpi 120 . . . . . . 7  |-  (EXMID  ->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
3219.21bi 1558 . . . . . 6  |-  (EXMID  ->  (
x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
4 df1o2 6432 . . . . . . . . 9  |-  1o  =  { (/) }
54pweqi 3581 . . . . . . . 8  |-  ~P 1o  =  ~P { (/) }
65eleq2i 2244 . . . . . . 7  |-  ( x  e.  ~P 1o  <->  x  e.  ~P { (/) } )
7 velpw 3584 . . . . . . 7  |-  ( x  e.  ~P { (/) }  <-> 
x  C_  { (/) } )
86, 7bitri 184 . . . . . 6  |-  ( x  e.  ~P 1o  <->  x  C_  { (/) } )
9 vex 2742 . . . . . . 7  |-  x  e. 
_V
109elpr 3615 . . . . . 6  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
113, 8, 103imtr4g 205 . . . . 5  |-  (EXMID  ->  (
x  e.  ~P 1o  ->  x  e.  { (/) ,  { (/) } } ) )
1211ssrdv 3163 . . . 4  |-  (EXMID  ->  ~P 1o  C_  { (/) ,  { (/)
} } )
13 pwpw0ss 3806 . . . . . 6  |-  { (/) ,  { (/) } }  C_  ~P { (/) }
1413, 5sseqtrri 3192 . . . . 5  |-  { (/) ,  { (/) } }  C_  ~P 1o
1514a1i 9 . . . 4  |-  (EXMID  ->  { (/) ,  { (/) } }  C_  ~P 1o )
1612, 15eqssd 3174 . . 3  |-  (EXMID  ->  ~P 1o  =  { (/) ,  { (/)
} } )
17 df2o2 6434 . . 3  |-  2o  =  { (/) ,  { (/) } }
1816, 17eqtr4di 2228 . 2  |-  (EXMID  ->  ~P 1o  =  2o )
19 simpr 110 . . . . . . . . 9  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  x  C_  { (/) } )
2019, 7sylibr 134 . . . . . . . 8  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  x  e.  ~P { (/) } )
2120, 5eleqtrrdi 2271 . . . . . . 7  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  x  e.  ~P 1o )
22 simpl 109 . . . . . . . 8  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  ~P 1o  =  2o )
2322, 17eqtrdi 2226 . . . . . . 7  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  ~P 1o  =  { (/) ,  { (/) } } )
2421, 23eleqtrd 2256 . . . . . 6  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  x  e.  {
(/) ,  { (/) } }
)
2524, 10sylib 122 . . . . 5  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  ( x  =  (/)  \/  x  =  { (/) } ) )
2625ex 115 . . . 4  |-  ( ~P 1o  =  2o  ->  ( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
2726alrimiv 1874 . . 3  |-  ( ~P 1o  =  2o  ->  A. x ( x  C_  {
(/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
2827, 1sylibr 134 . 2  |-  ( ~P 1o  =  2o  -> EXMID )
2918, 28impbii 126 1  |-  (EXMID  <->  ~P 1o  =  2o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708   A.wal 1351    = wceq 1353    e. wcel 2148    C_ wss 3131   (/)c0 3424   ~Pcpw 3577   {csn 3594   {cpr 3595  EXMIDwem 4196   1oc1o 6412   2oc2o 6413
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  ax-nul 4131
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-exmid 4197  df-suc 4373  df-1o 6419  df-2o 6420
This theorem is referenced by:  pw1fin  6912  pw1nel3  7232  3nsssucpw1  7237  onntri35  7238
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