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Theorem exmidn0m 4213
Description: Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)
Assertion
Ref Expression
exmidn0m  |-  (EXMID  <->  A. x
( x  =  (/)  \/ 
E. y  y  e.  x ) )
Distinct variable group:    x, y

Proof of Theorem exmidn0m
StepHypRef Expression
1 simpr 110 . . . . 5  |-  ( (EXMID  /\ 
E. y  y  e.  x )  ->  E. y 
y  e.  x )
21olcd 735 . . . 4  |-  ( (EXMID  /\ 
E. y  y  e.  x )  ->  (
x  =  (/)  \/  E. y  y  e.  x
) )
3 notm0 3455 . . . . . . 7  |-  ( -. 
E. y  y  e.  x  <->  x  =  (/) )
43biimpi 120 . . . . . 6  |-  ( -. 
E. y  y  e.  x  ->  x  =  (/) )
54adantl 277 . . . . 5  |-  ( (EXMID  /\ 
-.  E. y  y  e.  x )  ->  x  =  (/) )
65orcd 734 . . . 4  |-  ( (EXMID  /\ 
-.  E. y  y  e.  x )  ->  (
x  =  (/)  \/  E. y  y  e.  x
) )
7 exmidexmid 4208 . . . . 5  |-  (EXMID  -> DECID  E. y  y  e.  x )
8 exmiddc 837 . . . . 5  |-  (DECID  E. y 
y  e.  x  -> 
( E. y  y  e.  x  \/  -.  E. y  y  e.  x
) )
97, 8syl 14 . . . 4  |-  (EXMID  ->  ( E. y  y  e.  x  \/  -.  E. y 
y  e.  x ) )
102, 6, 9mpjaodan 799 . . 3  |-  (EXMID  ->  (
x  =  (/)  \/  E. y  y  e.  x
) )
1110alrimiv 1884 . 2  |-  (EXMID  ->  A. x
( x  =  (/)  \/ 
E. y  y  e.  x ) )
12 orc 713 . . . . . 6  |-  ( x  =  (/)  ->  ( x  =  (/)  \/  x  =  { (/) } ) )
1312a1d 22 . . . . 5  |-  ( x  =  (/)  ->  ( x 
C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
14 sssnm 3766 . . . . . . . 8  |-  ( E. y  y  e.  x  ->  ( x  C_  { (/) }  <-> 
x  =  { (/) } ) )
1514biimpa 296 . . . . . . 7  |-  ( ( E. y  y  e.  x  /\  x  C_  {
(/) } )  ->  x  =  { (/) } )
1615olcd 735 . . . . . 6  |-  ( ( E. y  y  e.  x  /\  x  C_  {
(/) } )  ->  (
x  =  (/)  \/  x  =  { (/) } ) )
1716ex 115 . . . . 5  |-  ( E. y  y  e.  x  ->  ( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
1813, 17jaoi 717 . . . 4  |-  ( ( x  =  (/)  \/  E. y  y  e.  x
)  ->  ( x  C_ 
{ (/) }  ->  (
x  =  (/)  \/  x  =  { (/) } ) ) )
1918alimi 1465 . . 3  |-  ( A. x ( x  =  (/)  \/  E. y  y  e.  x )  ->  A. x ( x  C_  {
(/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
20 exmid01 4210 . . 3  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
2119, 20sylibr 134 . 2  |-  ( A. x ( x  =  (/)  \/  E. y  y  e.  x )  -> EXMID )
2211, 21impbii 126 1  |-  (EXMID  <->  A. x
( x  =  (/)  \/ 
E. y  y  e.  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709  DECID wdc 835   A.wal 1361    = wceq 1363   E.wex 1502    C_ wss 3141   (/)c0 3434   {csn 3604  EXMIDwem 4206
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-v 2751  df-dif 3143  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-exmid 4207
This theorem is referenced by:  exmidsssn  4214
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