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Theorem exmidn0m 4062
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 109 . . . . 5  |-  ( (EXMID  /\ 
E. y  y  e.  x )  ->  E. y 
y  e.  x )
21olcd 694 . . . 4  |-  ( (EXMID  /\ 
E. y  y  e.  x )  ->  (
x  =  (/)  \/  E. y  y  e.  x
) )
3 notm0 3330 . . . . . . 7  |-  ( -. 
E. y  y  e.  x  <->  x  =  (/) )
43biimpi 119 . . . . . 6  |-  ( -. 
E. y  y  e.  x  ->  x  =  (/) )
54adantl 273 . . . . 5  |-  ( (EXMID  /\ 
-.  E. y  y  e.  x )  ->  x  =  (/) )
65orcd 693 . . . 4  |-  ( (EXMID  /\ 
-.  E. y  y  e.  x )  ->  (
x  =  (/)  \/  E. y  y  e.  x
) )
7 exmidexmid 4060 . . . . 5  |-  (EXMID  -> DECID  E. y  y  e.  x )
8 exmiddc 788 . . . . 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 753 . . 3  |-  (EXMID  ->  (
x  =  (/)  \/  E. y  y  e.  x
) )
1110alrimiv 1813 . 2  |-  (EXMID  ->  A. x
( x  =  (/)  \/ 
E. y  y  e.  x ) )
12 orc 674 . . . . . 6  |-  ( x  =  (/)  ->  ( x  =  (/)  \/  x  =  { (/) } ) )
1312a1d 22 . . . . 5  |-  ( x  =  (/)  ->  ( x 
C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
14 sssnm 3628 . . . . . . . 8  |-  ( E. y  y  e.  x  ->  ( x  C_  { (/) }  <-> 
x  =  { (/) } ) )
1514biimpa 292 . . . . . . 7  |-  ( ( E. y  y  e.  x  /\  x  C_  {
(/) } )  ->  x  =  { (/) } )
1615olcd 694 . . . . . 6  |-  ( ( E. y  y  e.  x  /\  x  C_  {
(/) } )  ->  (
x  =  (/)  \/  x  =  { (/) } ) )
1716ex 114 . . . . 5  |-  ( E. y  y  e.  x  ->  ( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
1813, 17jaoi 677 . . . 4  |-  ( ( x  =  (/)  \/  E. y  y  e.  x
)  ->  ( x  C_ 
{ (/) }  ->  (
x  =  (/)  \/  x  =  { (/) } ) ) )
1918alimi 1399 . . 3  |-  ( A. x ( x  =  (/)  \/  E. y  y  e.  x )  ->  A. x ( x  C_  {
(/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
20 exmid01 4061 . . 3  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
2119, 20sylibr 133 . 2  |-  ( A. x ( x  =  (/)  \/  E. y  y  e.  x )  -> EXMID )
2211, 21impbii 125 1  |-  (EXMID  <->  A. x
( x  =  (/)  \/ 
E. y  y  e.  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 670  DECID wdc 786   A.wal 1297    = wceq 1299   E.wex 1436    C_ wss 3021   (/)c0 3310   {csn 3474  EXMIDwem 4058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038
This theorem depends on definitions:  df-bi 116  df-dc 787  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rab 2384  df-v 2643  df-dif 3023  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-exmid 4059
This theorem is referenced by:  exmidsssn  4063
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