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Theorem exmidpw 6886
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 6408 . . . . 5  |-  1o  =  { (/) }
2 p0ex 4174 . . . . 5  |-  { (/) }  e.  _V
31, 2eqeltri 2243 . . . 4  |-  1o  e.  _V
43pwex 4169 . . 3  |-  ~P 1o  e.  _V
5 exmid01 4184 . . . . . . . . 9  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
65biimpi 119 . . . . . . . 8  |-  (EXMID  ->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
7619.21bi 1551 . . . . . . 7  |-  (EXMID  ->  (
x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
81pweqi 3570 . . . . . . . . 9  |-  ~P 1o  =  ~P { (/) }
98eleq2i 2237 . . . . . . . 8  |-  ( x  e.  ~P 1o  <->  x  e.  ~P { (/) } )
10 velpw 3573 . . . . . . . 8  |-  ( x  e.  ~P { (/) }  <-> 
x  C_  { (/) } )
119, 10bitri 183 . . . . . . 7  |-  ( x  e.  ~P 1o  <->  x  C_  { (/) } )
12 vex 2733 . . . . . . . 8  |-  x  e. 
_V
1312elpr 3604 . . . . . . 7  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
147, 11, 133imtr4g 204 . . . . . 6  |-  (EXMID  ->  (
x  e.  ~P 1o  ->  x  e.  { (/) ,  { (/) } } ) )
1514ssrdv 3153 . . . . 5  |-  (EXMID  ->  ~P 1o  C_  { (/) ,  { (/)
} } )
16 pwpw0ss 3791 . . . . . . 7  |-  { (/) ,  { (/) } }  C_  ~P { (/) }
1716, 8sseqtrri 3182 . . . . . 6  |-  { (/) ,  { (/) } }  C_  ~P 1o
1817a1i 9 . . . . 5  |-  (EXMID  ->  { (/) ,  { (/) } }  C_  ~P 1o )
1915, 18eqssd 3164 . . . 4  |-  (EXMID  ->  ~P 1o  =  { (/) ,  { (/)
} } )
20 df2o2 6410 . . . 4  |-  2o  =  { (/) ,  { (/) } }
2119, 20eqtr4di 2221 . . 3  |-  (EXMID  ->  ~P 1o  =  2o )
22 eqeng 6744 . . 3  |-  ( ~P 1o  e.  _V  ->  ( ~P 1o  =  2o 
->  ~P 1o  ~~  2o ) )
234, 21, 22mpsyl 65 . 2  |-  (EXMID  ->  ~P 1o  ~~  2o )
24 0nep0 4151 . . . . . . . 8  |-  (/)  =/=  { (/)
}
25 0ex 4116 . . . . . . . . . . 11  |-  (/)  e.  _V
2625, 2prss 3736 . . . . . . . . . 10  |-  ( (
(/)  e.  ~P 1o  /\ 
{ (/) }  e.  ~P 1o )  <->  { (/) ,  { (/) } }  C_  ~P 1o )
2717, 26mpbir 145 . . . . . . . . 9  |-  ( (/)  e.  ~P 1o  /\  { (/)
}  e.  ~P 1o )
28 en2eqpr 6885 . . . . . . . . . 10  |-  ( ( ~P 1o  ~~  2o  /\  (/)  e.  ~P 1o  /\  {
(/) }  e.  ~P 1o )  ->  ( (/)  =/=  { (/) }  ->  ~P 1o  =  { (/) ,  { (/)
} } ) )
29283expb 1199 . . . . . . . . 9  |-  ( ( ~P 1o  ~~  2o  /\  ( (/)  e.  ~P 1o  /\  { (/) }  e.  ~P 1o ) )  -> 
( (/)  =/=  { (/) }  ->  ~P 1o  =  { (/) ,  { (/) } } ) )
3027, 29mpan2 423 . . . . . . . 8  |-  ( ~P 1o  ~~  2o  ->  (
(/)  =/=  { (/) }  ->  ~P 1o  =  { (/) ,  { (/) } } ) )
3124, 30mpi 15 . . . . . . 7  |-  ( ~P 1o  ~~  2o  ->  ~P 1o  =  { (/) ,  { (/) } } )
3231eleq2d 2240 . . . . . 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 1867 . . 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 703   A.wal 1346    = wceq 1348    e. wcel 2141    =/= wne 2340   _Vcvv 2730    C_ wss 3121   (/)c0 3414   ~Pcpw 3566   {csn 3583   {cpr 3584   class class class wbr 3989  EXMIDwem 4180   1oc1o 6388   2oc2o 6389    ~~ cen 6716
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-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  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-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-exmid 4181  df-id 4278  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1o 6395  df-2o 6396  df-en 6719
This theorem is referenced by:  pwf1oexmid  14032
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