ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pw1dc0el Unicode version

Theorem pw1dc0el 6889
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 6408 . . . . . . 7  |-  1o  =  { (/) }
21eqcomi 2174 . . . . . 6  |-  { (/) }  =  1o
32sseq2i 3174 . . . . 5  |-  ( x 
C_  { (/) }  <->  x  C_  1o )
4 velpw 3573 . . . . 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 1463 . 2  |-  ( A. x ( x  C_  {
(/) }  -> DECID  (/)  e.  x )  <->  A. x ( x  e. 
~P 1o  -> DECID  (/)  e.  x ) )
8 df-exmid 4181 . 2  |-  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
9 df-ral 2453 . 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 829   A.wal 1346    e. wcel 2141   A.wral 2448    C_ wss 3121   (/)c0 3414   ~Pcpw 3566   {csn 3583  EXMIDwem 4180   1oc1o 6388
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-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-exmid 4181  df-suc 4356  df-1o 6395
This theorem is referenced by:  pw1dc1  6891
  Copyright terms: Public domain W3C validator