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Theorem reg3exmid 4597
Description: If any inhabited set satisfying df-wetr 4352 for  _E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
Hypothesis
Ref Expression
reg3exmid.1  |-  ( (  _E  We  z  /\  E. w  w  e.  z )  ->  E. x  e.  z  A. y  e.  z  x  C_  y
)
Assertion
Ref Expression
reg3exmid  |-  ( ph  \/  -.  ph )
Distinct variable groups:    ph, w, z    ph, x, y, z

Proof of Theorem reg3exmid
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 eqid 2189 . . 3  |-  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  =  { u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) }
21regexmidlemm 4549 . 2  |-  E. w  w  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }
31reg3exmidlemwe 4596 . . 3  |-  _E  We  { u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) }
4 pp0ex 4207 . . . . 5  |-  { (/) ,  { (/) } }  e.  _V
54rabex 4162 . . . 4  |-  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  e.  _V
6 weeq2 4375 . . . . . 6  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  (  _E  We  z  <->  _E  We  { u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) } ) )
7 eleq2 2253 . . . . . . 7  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  ( w  e.  z  <->  w  e.  { u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) } ) )
87exbidv 1836 . . . . . 6  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  ( E. w  w  e.  z  <->  E. w  w  e. 
{ u  e.  { (/)
,  { (/) } }  |  ( u  =  { (/) }  \/  (
u  =  (/)  /\  ph ) ) } ) )
96, 8anbi12d 473 . . . . 5  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  ( (  _E  We  z  /\  E. w  w  e.  z )  <->  (  _E  We  { u  e.  { (/)
,  { (/) } }  |  ( u  =  { (/) }  \/  (
u  =  (/)  /\  ph ) ) }  /\  E. w  w  e.  {
u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) } ) ) )
10 raleq 2686 . . . . . 6  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  ( A. y  e.  z  x  C_  y  <->  A. y  e.  { u  e.  { (/)
,  { (/) } }  |  ( u  =  { (/) }  \/  (
u  =  (/)  /\  ph ) ) } x  C_  y ) )
1110rexeqbi1dv 2695 . . . . 5  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  ( E. x  e.  z 
A. y  e.  z  x  C_  y  <->  E. x  e.  { u  e.  { (/)
,  { (/) } }  |  ( u  =  { (/) }  \/  (
u  =  (/)  /\  ph ) ) } A. y  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) } x  C_  y ) )
129, 11imbi12d 234 . . . 4  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  ( ( (  _E  We  z  /\  E. w  w  e.  z )  ->  E. x  e.  z  A. y  e.  z  x  C_  y )  <->  ( (  _E  We  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  /\  E. w  w  e.  {
u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) } )  ->  E. x  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) } A. y  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) } x  C_  y ) ) )
13 reg3exmid.1 . . . 4  |-  ( (  _E  We  z  /\  E. w  w  e.  z )  ->  E. x  e.  z  A. y  e.  z  x  C_  y
)
145, 12, 13vtocl 2806 . . 3  |-  ( (  _E  We  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  /\  E. w  w  e.  {
u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) } )  ->  E. x  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) } A. y  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) } x  C_  y )
153, 14mpan 424 . 2  |-  ( E. w  w  e.  {
u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) }  ->  E. x  e.  { u  e.  { (/)
,  { (/) } }  |  ( u  =  { (/) }  \/  (
u  =  (/)  /\  ph ) ) } A. y  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) } x  C_  y )
161reg2exmidlema 4551 . 2  |-  ( E. x  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) } A. y  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) } x  C_  y  ->  ( ph  \/  -.  ph ) )
172, 15, 16mp2b 8 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 709    = wceq 1364   E.wex 1503    e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472    C_ wss 3144   (/)c0 3437   {csn 3607   {cpr 3608    _E cep 4305    We wwe 4348
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-eprel 4307  df-frfor 4349  df-frind 4350  df-wetr 4352
This theorem is referenced by: (None)
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