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Theorem pw1dc0el 6853
Description: Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.)
Assertion
Ref Expression
pw1dc0el  |-  (EXMID  <->  A. x  e.  ~P  1oDECID  (/)  e.  x )

Proof of Theorem pw1dc0el
StepHypRef Expression
1 df1o2 6373 . . . . . . 7  |-  1o  =  { (/) }
21eqcomi 2161 . . . . . 6  |-  { (/) }  =  1o
32sseq2i 3155 . . . . 5  |-  ( x 
C_  { (/) }  <->  x  C_  1o )
4 velpw 3550 . . . . 5  |-  ( x  e.  ~P 1o  <->  x  C_  1o )
53, 4bitr4i 186 . . . 4  |-  ( x 
C_  { (/) }  <->  x  e.  ~P 1o )
65imbi1i 237 . . 3  |-  ( ( x  C_  { (/) }  -> DECID  (/)  e.  x
)  <->  ( x  e. 
~P 1o  -> DECID  (/)  e.  x ) )
76albii 1450 . 2  |-  ( A. x ( x  C_  {
(/) }  -> DECID  (/)  e.  x )  <->  A. x ( x  e. 
~P 1o  -> DECID  (/)  e.  x ) )
8 df-exmid 4156 . 2  |-  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
9 df-ral 2440 . 2  |-  ( A. x  e.  ~P  1oDECID  (/)  e.  x  <->  A. x ( x  e. 
~P 1o  -> DECID  (/)  e.  x ) )
107, 8, 93bitr4i 211 1  |-  (EXMID  <->  A. x  e.  ~P  1oDECID  (/)  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104  DECID wdc 820   A.wal 1333    e. wcel 2128   A.wral 2435    C_ wss 3102   (/)c0 3394   ~Pcpw 3543   {csn 3560  EXMIDwem 4155   1oc1o 6353
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
This theorem depends on definitions:  df-bi 116  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-exmid 4156  df-suc 4331  df-1o 6360
This theorem is referenced by:  pw1dc1  6855
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