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Theorem pw1dc1 6859
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 6857 . 2  |-  (EXMID  <->  A. x  e.  ~P  1oDECID  (/)  e.  x )
2 elpwi 3552 . . . . 5  |-  ( x  e.  ~P 1o  ->  x 
C_  1o )
3 ss1o0el1o 6858 . . . . 5  |-  ( x 
C_  1o  ->  ( (/)  e.  x  <->  x  =  1o ) )
42, 3syl 14 . . . 4  |-  ( x  e.  ~P 1o  ->  (
(/)  e.  x  <->  x  =  1o ) )
54dcbid 824 . . 3  |-  ( x  e.  ~P 1o  ->  (DECID  (/)  e.  x  <-> DECID  x  =  1o )
)
65ralbiia 2471 . 2  |-  ( A. x  e.  ~P  1oDECID  (/)  e.  x  <->  A. x  e.  ~P  1oDECID  x  =  1o )
71, 6bitri 183 1  |-  (EXMID  <->  A. x  e.  ~P  1oDECID  x  =  1o )
Colors of variables: wff set class
Syntax hints:    <-> wb 104  DECID wdc 820    = wceq 1335    e. wcel 2128   A.wral 2435    C_ wss 3102   (/)c0 3394   ~Pcpw 3543  EXMIDwem 4156   1oc1o 6357
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-ral 2440  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-exmid 4157  df-suc 4332  df-1o 6364
This theorem is referenced by: (None)
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