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Theorem exmidpweq 6855
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 4160 . . . . . . . 8  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
21biimpi 119 . . . . . . 7  |-  (EXMID  ->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
3219.21bi 1538 . . . . . 6  |-  (EXMID  ->  (
x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
4 df1o2 6377 . . . . . . . . 9  |-  1o  =  { (/) }
54pweqi 3547 . . . . . . . 8  |-  ~P 1o  =  ~P { (/) }
65eleq2i 2224 . . . . . . 7  |-  ( x  e.  ~P 1o  <->  x  e.  ~P { (/) } )
7 velpw 3550 . . . . . . 7  |-  ( x  e.  ~P { (/) }  <-> 
x  C_  { (/) } )
86, 7bitri 183 . . . . . 6  |-  ( x  e.  ~P 1o  <->  x  C_  { (/) } )
9 vex 2715 . . . . . . 7  |-  x  e. 
_V
109elpr 3581 . . . . . 6  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
113, 8, 103imtr4g 204 . . . . 5  |-  (EXMID  ->  (
x  e.  ~P 1o  ->  x  e.  { (/) ,  { (/) } } ) )
1211ssrdv 3134 . . . 4  |-  (EXMID  ->  ~P 1o  C_  { (/) ,  { (/)
} } )
13 pwpw0ss 3768 . . . . . 6  |-  { (/) ,  { (/) } }  C_  ~P { (/) }
1413, 5sseqtrri 3163 . . . . 5  |-  { (/) ,  { (/) } }  C_  ~P 1o
1514a1i 9 . . . 4  |-  (EXMID  ->  { (/) ,  { (/) } }  C_  ~P 1o )
1612, 15eqssd 3145 . . 3  |-  (EXMID  ->  ~P 1o  =  { (/) ,  { (/)
} } )
17 df2o2 6379 . . 3  |-  2o  =  { (/) ,  { (/) } }
1816, 17eqtr4di 2208 . 2  |-  (EXMID  ->  ~P 1o  =  2o )
19 simpr 109 . . . . . . . . 9  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  x  C_  { (/) } )
2019, 7sylibr 133 . . . . . . . 8  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  x  e.  ~P { (/) } )
2120, 5eleqtrrdi 2251 . . . . . . 7  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  x  e.  ~P 1o )
22 simpl 108 . . . . . . . 8  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  ~P 1o  =  2o )
2322, 17eqtrdi 2206 . . . . . . 7  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  ~P 1o  =  { (/) ,  { (/) } } )
2421, 23eleqtrd 2236 . . . . . 6  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  x  e.  {
(/) ,  { (/) } }
)
2524, 10sylib 121 . . . . 5  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  ( x  =  (/)  \/  x  =  { (/) } ) )
2625ex 114 . . . 4  |-  ( ~P 1o  =  2o  ->  ( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
2726alrimiv 1854 . . 3  |-  ( ~P 1o  =  2o  ->  A. x ( x  C_  {
(/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
2827, 1sylibr 133 . 2  |-  ( ~P 1o  =  2o  -> EXMID )
2918, 28impbii 125 1  |-  (EXMID  <->  ~P 1o  =  2o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698   A.wal 1333    = wceq 1335    e. wcel 2128    C_ wss 3102   (/)c0 3394   ~Pcpw 3543   {csn 3560   {cpr 3561  EXMIDwem 4156   1oc1o 6357   2oc2o 6358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-nul 4091
This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-exmid 4157  df-suc 4332  df-1o 6364  df-2o 6365
This theorem is referenced by:  pw1fin  6856  pw1nel3  7167  3nsssucpw1  7172  onntri35  7173
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