ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reg3exmid Unicode version

Theorem reg3exmid 4462
Description: If any inhabited set satisfying df-wetr 4224 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 2115 . . 3  |-  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  =  { u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) }
21regexmidlemm 4415 . 2  |-  E. w  w  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }
31reg3exmidlemwe 4461 . . 3  |-  _E  We  { u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) }
4 pp0ex 4081 . . . . 5  |-  { (/) ,  { (/) } }  e.  _V
54rabex 4040 . . . 4  |-  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  e.  _V
6 weeq2 4247 . . . . . 6  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  (  _E  We  z  <->  _E  We  { u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) } ) )
7 eleq2 2179 . . . . . . 7  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  ( w  e.  z  <->  w  e.  { u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) } ) )
87exbidv 1779 . . . . . 6  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  ( E. w  w  e.  z  <->  E. w  w  e. 
{ u  e.  { (/)
,  { (/) } }  |  ( u  =  { (/) }  \/  (
u  =  (/)  /\  ph ) ) } ) )
96, 8anbi12d 462 . . . . 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 2601 . . . . . 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 2610 . . . . 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 233 . . . 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 2712 . . 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 418 . 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 4417 . 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 103    \/ wo 680    = wceq 1314   E.wex 1451    e. wcel 1463   A.wral 2391   E.wrex 2392   {crab 2395    C_ wss 3039   (/)c0 3331   {csn 3495   {cpr 3496    _E cep 4177    We wwe 4220
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-eprel 4179  df-frfor 4221  df-frind 4222  df-wetr 4224
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator