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Theorem exmidn0m 4185
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 729 . . . 4  |-  ( (EXMID  /\ 
E. y  y  e.  x )  ->  (
x  =  (/)  \/  E. y  y  e.  x
) )
3 notm0 3434 . . . . . . 7  |-  ( -. 
E. y  y  e.  x  <->  x  =  (/) )
43biimpi 119 . . . . . 6  |-  ( -. 
E. y  y  e.  x  ->  x  =  (/) )
54adantl 275 . . . . 5  |-  ( (EXMID  /\ 
-.  E. y  y  e.  x )  ->  x  =  (/) )
65orcd 728 . . . 4  |-  ( (EXMID  /\ 
-.  E. y  y  e.  x )  ->  (
x  =  (/)  \/  E. y  y  e.  x
) )
7 exmidexmid 4180 . . . . 5  |-  (EXMID  -> DECID  E. y  y  e.  x )
8 exmiddc 831 . . . . 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 793 . . 3  |-  (EXMID  ->  (
x  =  (/)  \/  E. y  y  e.  x
) )
1110alrimiv 1867 . 2  |-  (EXMID  ->  A. x
( x  =  (/)  \/ 
E. y  y  e.  x ) )
12 orc 707 . . . . . 6  |-  ( x  =  (/)  ->  ( x  =  (/)  \/  x  =  { (/) } ) )
1312a1d 22 . . . . 5  |-  ( x  =  (/)  ->  ( x 
C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
14 sssnm 3739 . . . . . . . 8  |-  ( E. y  y  e.  x  ->  ( x  C_  { (/) }  <-> 
x  =  { (/) } ) )
1514biimpa 294 . . . . . . 7  |-  ( ( E. y  y  e.  x  /\  x  C_  {
(/) } )  ->  x  =  { (/) } )
1615olcd 729 . . . . . 6  |-  ( ( E. y  y  e.  x  /\  x  C_  {
(/) } )  ->  (
x  =  (/)  \/  x  =  { (/) } ) )
1716ex 114 . . . . 5  |-  ( E. y  y  e.  x  ->  ( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
1813, 17jaoi 711 . . . 4  |-  ( ( x  =  (/)  \/  E. y  y  e.  x
)  ->  ( x  C_ 
{ (/) }  ->  (
x  =  (/)  \/  x  =  { (/) } ) ) )
1918alimi 1448 . . 3  |-  ( A. x ( x  =  (/)  \/  E. y  y  e.  x )  ->  A. x ( x  C_  {
(/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
20 exmid01 4182 . . 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 703  DECID wdc 829   A.wal 1346    = wceq 1348   E.wex 1485    C_ wss 3121   (/)c0 3414   {csn 3581  EXMIDwem 4178
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-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-exmid 4179
This theorem is referenced by:  exmidsssn  4186
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