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Theorem pw1dc1 6910
Description: If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.)
Assertion
Ref Expression
pw1dc1  |-  (EXMID  <->  A. x  e.  ~P  1oDECID  x  =  1o )

Proof of Theorem pw1dc1
StepHypRef Expression
1 pw1dc0el 6908 . 2  |-  (EXMID  <->  A. x  e.  ~P  1oDECID  (/)  e.  x )
2 elpwi 3584 . . . . 5  |-  ( x  e.  ~P 1o  ->  x 
C_  1o )
3 ss1o0el1o 6909 . . . . 5  |-  ( x 
C_  1o  ->  ( (/)  e.  x  <->  x  =  1o ) )
42, 3syl 14 . . . 4  |-  ( x  e.  ~P 1o  ->  (
(/)  e.  x  <->  x  =  1o ) )
54dcbid 838 . . 3  |-  ( x  e.  ~P 1o  ->  (DECID  (/)  e.  x  <-> DECID  x  =  1o )
)
65ralbiia 2491 . 2  |-  ( A. x  e.  ~P  1oDECID  (/)  e.  x  <->  A. x  e.  ~P  1oDECID  x  =  1o )
71, 6bitri 184 1  |-  (EXMID  <->  A. x  e.  ~P  1oDECID  x  =  1o )
Colors of variables: wff set class
Syntax hints:    <-> wb 105  DECID wdc 834    = wceq 1353    e. wcel 2148   A.wral 2455    C_ wss 3129   (/)c0 3422   ~Pcpw 3575  EXMIDwem 4193   1oc1o 6407
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 4128
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-ral 2460  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-exmid 4194  df-suc 4370  df-1o 6414
This theorem is referenced by: (None)
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