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Theorem exmidpweq 6887
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 4184 . . . . . . . 8  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
21biimpi 119 . . . . . . 7  |-  (EXMID  ->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
3219.21bi 1551 . . . . . 6  |-  (EXMID  ->  (
x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
4 df1o2 6408 . . . . . . . . 9  |-  1o  =  { (/) }
54pweqi 3570 . . . . . . . 8  |-  ~P 1o  =  ~P { (/) }
65eleq2i 2237 . . . . . . 7  |-  ( x  e.  ~P 1o  <->  x  e.  ~P { (/) } )
7 velpw 3573 . . . . . . 7  |-  ( x  e.  ~P { (/) }  <-> 
x  C_  { (/) } )
86, 7bitri 183 . . . . . 6  |-  ( x  e.  ~P 1o  <->  x  C_  { (/) } )
9 vex 2733 . . . . . . 7  |-  x  e. 
_V
109elpr 3604 . . . . . 6  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
113, 8, 103imtr4g 204 . . . . 5  |-  (EXMID  ->  (
x  e.  ~P 1o  ->  x  e.  { (/) ,  { (/) } } ) )
1211ssrdv 3153 . . . 4  |-  (EXMID  ->  ~P 1o  C_  { (/) ,  { (/)
} } )
13 pwpw0ss 3791 . . . . . 6  |-  { (/) ,  { (/) } }  C_  ~P { (/) }
1413, 5sseqtrri 3182 . . . . 5  |-  { (/) ,  { (/) } }  C_  ~P 1o
1514a1i 9 . . . 4  |-  (EXMID  ->  { (/) ,  { (/) } }  C_  ~P 1o )
1612, 15eqssd 3164 . . 3  |-  (EXMID  ->  ~P 1o  =  { (/) ,  { (/)
} } )
17 df2o2 6410 . . 3  |-  2o  =  { (/) ,  { (/) } }
1816, 17eqtr4di 2221 . 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 2264 . . . . . . 7  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  x  e.  ~P 1o )
22 simpl 108 . . . . . . . 8  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  ~P 1o  =  2o )
2322, 17eqtrdi 2219 . . . . . . 7  |-  ( ( ~P 1o  =  2o 
/\  x  C_  { (/) } )  ->  ~P 1o  =  { (/) ,  { (/) } } )
2421, 23eleqtrd 2249 . . . . . 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 1867 . . 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 703   A.wal 1346    = wceq 1348    e. wcel 2141    C_ wss 3121   (/)c0 3414   ~Pcpw 3566   {csn 3583   {cpr 3584  EXMIDwem 4180   1oc1o 6388   2oc2o 6389
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-exmid 4181  df-suc 4356  df-1o 6395  df-2o 6396
This theorem is referenced by:  pw1fin  6888  pw1nel3  7208  3nsssucpw1  7213  onntri35  7214
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