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Theorem pw1dc0el 6910
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 6429 . . . . . . 7  |-  1o  =  { (/) }
21eqcomi 2181 . . . . . 6  |-  { (/) }  =  1o
32sseq2i 3182 . . . . 5  |-  ( x 
C_  { (/) }  <->  x  C_  1o )
4 velpw 3582 . . . . 5  |-  ( x  e.  ~P 1o  <->  x  C_  1o )
53, 4bitr4i 187 . . . 4  |-  ( x 
C_  { (/) }  <->  x  e.  ~P 1o )
65imbi1i 238 . . 3  |-  ( ( x  C_  { (/) }  -> DECID  (/)  e.  x
)  <->  ( x  e. 
~P 1o  -> DECID  (/)  e.  x ) )
76albii 1470 . 2  |-  ( A. x ( x  C_  {
(/) }  -> DECID  (/)  e.  x )  <->  A. x ( x  e. 
~P 1o  -> DECID  (/)  e.  x ) )
8 df-exmid 4195 . 2  |-  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
9 df-ral 2460 . 2  |-  ( A. x  e.  ~P  1oDECID  (/)  e.  x  <->  A. x ( x  e. 
~P 1o  -> DECID  (/)  e.  x ) )
107, 8, 93bitr4i 212 1  |-  (EXMID  <->  A. x  e.  ~P  1oDECID  (/)  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105  DECID wdc 834   A.wal 1351    e. wcel 2148   A.wral 2455    C_ wss 3129   (/)c0 3422   ~Pcpw 3575   {csn 3592  EXMIDwem 4194   1oc1o 6409
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
This theorem depends on definitions:  df-bi 117  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-exmid 4195  df-suc 4371  df-1o 6416
This theorem is referenced by:  pw1dc1  6912
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