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Theorem exmidpw 6809
Description: Excluded middle is equivalent to the power set of  1o having two elements. Remark of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 30-Jun-2022.)
Assertion
Ref Expression
exmidpw  |-  (EXMID  <->  ~P 1o  ~~  2o )

Proof of Theorem exmidpw
StepHypRef Expression
1 df1o2 6333 . . . . 5  |-  1o  =  { (/) }
2 p0ex 4119 . . . . 5  |-  { (/) }  e.  _V
31, 2eqeltri 2213 . . . 4  |-  1o  e.  _V
43pwex 4114 . . 3  |-  ~P 1o  e.  _V
5 exmid01 4128 . . . . . . . . 9  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
65biimpi 119 . . . . . . . 8  |-  (EXMID  ->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
7619.21bi 1538 . . . . . . 7  |-  (EXMID  ->  (
x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
81pweqi 3518 . . . . . . . . 9  |-  ~P 1o  =  ~P { (/) }
98eleq2i 2207 . . . . . . . 8  |-  ( x  e.  ~P 1o  <->  x  e.  ~P { (/) } )
10 velpw 3521 . . . . . . . 8  |-  ( x  e.  ~P { (/) }  <-> 
x  C_  { (/) } )
119, 10bitri 183 . . . . . . 7  |-  ( x  e.  ~P 1o  <->  x  C_  { (/) } )
12 vex 2692 . . . . . . . 8  |-  x  e. 
_V
1312elpr 3552 . . . . . . 7  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
147, 11, 133imtr4g 204 . . . . . 6  |-  (EXMID  ->  (
x  e.  ~P 1o  ->  x  e.  { (/) ,  { (/) } } ) )
1514ssrdv 3107 . . . . 5  |-  (EXMID  ->  ~P 1o  C_  { (/) ,  { (/)
} } )
16 pwpw0ss 3738 . . . . . . 7  |-  { (/) ,  { (/) } }  C_  ~P { (/) }
1716, 8sseqtrri 3136 . . . . . 6  |-  { (/) ,  { (/) } }  C_  ~P 1o
1817a1i 9 . . . . 5  |-  (EXMID  ->  { (/) ,  { (/) } }  C_  ~P 1o )
1915, 18eqssd 3118 . . . 4  |-  (EXMID  ->  ~P 1o  =  { (/) ,  { (/)
} } )
20 df2o2 6335 . . . 4  |-  2o  =  { (/) ,  { (/) } }
2119, 20eqtr4di 2191 . . 3  |-  (EXMID  ->  ~P 1o  =  2o )
22 eqeng 6667 . . 3  |-  ( ~P 1o  e.  _V  ->  ( ~P 1o  =  2o 
->  ~P 1o  ~~  2o ) )
234, 21, 22mpsyl 65 . 2  |-  (EXMID  ->  ~P 1o  ~~  2o )
24 0nep0 4096 . . . . . . . 8  |-  (/)  =/=  { (/)
}
25 0ex 4062 . . . . . . . . . . 11  |-  (/)  e.  _V
2625, 2prss 3683 . . . . . . . . . 10  |-  ( (
(/)  e.  ~P 1o  /\ 
{ (/) }  e.  ~P 1o )  <->  { (/) ,  { (/) } }  C_  ~P 1o )
2717, 26mpbir 145 . . . . . . . . 9  |-  ( (/)  e.  ~P 1o  /\  { (/)
}  e.  ~P 1o )
28 en2eqpr 6808 . . . . . . . . . 10  |-  ( ( ~P 1o  ~~  2o  /\  (/)  e.  ~P 1o  /\  {
(/) }  e.  ~P 1o )  ->  ( (/)  =/=  { (/) }  ->  ~P 1o  =  { (/) ,  { (/)
} } ) )
29283expb 1183 . . . . . . . . 9  |-  ( ( ~P 1o  ~~  2o  /\  ( (/)  e.  ~P 1o  /\  { (/) }  e.  ~P 1o ) )  -> 
( (/)  =/=  { (/) }  ->  ~P 1o  =  { (/) ,  { (/) } } ) )
3027, 29mpan2 422 . . . . . . . 8  |-  ( ~P 1o  ~~  2o  ->  (
(/)  =/=  { (/) }  ->  ~P 1o  =  { (/) ,  { (/) } } ) )
3124, 30mpi 15 . . . . . . 7  |-  ( ~P 1o  ~~  2o  ->  ~P 1o  =  { (/) ,  { (/) } } )
3231eleq2d 2210 . . . . . 6  |-  ( ~P 1o  ~~  2o  ->  ( x  e.  ~P 1o  <->  x  e.  { (/) ,  { (/)
} } ) )
3332, 11, 133bitr3g 221 . . . . 5  |-  ( ~P 1o  ~~  2o  ->  ( x  C_  { (/) }  <->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
3433biimpd 143 . . . 4  |-  ( ~P 1o  ~~  2o  ->  ( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
3534alrimiv 1847 . . 3  |-  ( ~P 1o  ~~  2o  ->  A. x ( x  C_  {
(/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
3635, 5sylibr 133 . 2  |-  ( ~P 1o  ~~  2o  -> EXMID )
3723, 36impbii 125 1  |-  (EXMID  <->  ~P 1o  ~~  2o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698   A.wal 1330    = wceq 1332    e. wcel 1481    =/= wne 2309   _Vcvv 2689    C_ wss 3075   (/)c0 3367   ~Pcpw 3514   {csn 3531   {cpr 3532   class class class wbr 3936  EXMIDwem 4125   1oc1o 6313   2oc2o 6314    ~~ cen 6639
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-exmid 4126  df-id 4222  df-suc 4300  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-1o 6320  df-2o 6321  df-en 6642
This theorem is referenced by:  pwf1oexmid  13365
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