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Theorem reg2exmid 4289
Description: If any inhabited set has a minimal element (when expressed by  C_), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
Hypothesis
Ref Expression
reg2exmid.1  |-  A. z
( E. w  w  e.  z  ->  E. x  e.  z  A. y  e.  z  x  C_  y
)
Assertion
Ref Expression
reg2exmid  |-  ( ph  \/  -.  ph )
Distinct variable groups:    ph, w, z    ph, x, z, y

Proof of Theorem reg2exmid
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 eqid 2056 . . . 4  |-  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  =  { u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) }
21regexmidlemm 4285 . . 3  |-  E. w  w  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }
3 reg2exmid.1 . . . 4  |-  A. z
( E. w  w  e.  z  ->  E. x  e.  z  A. y  e.  z  x  C_  y
)
4 pp0ex 3968 . . . . . 6  |-  { (/) ,  { (/) } }  e.  _V
54rabex 3929 . . . . 5  |-  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  e.  _V
6 eleq2 2117 . . . . . . 7  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  ( w  e.  z  <->  w  e.  { u  e.  { (/) ,  { (/) } }  | 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) } ) )
76exbidv 1722 . . . . . 6  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  ( E. w  w  e.  z  <->  E. w  w  e. 
{ u  e.  { (/)
,  { (/) } }  |  ( u  =  { (/) }  \/  (
u  =  (/)  /\  ph ) ) } ) )
8 raleq 2522 . . . . . . 7  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  ( A. y  e.  z  x  C_  y  <->  A. y  e.  { u  e.  { (/)
,  { (/) } }  |  ( u  =  { (/) }  \/  (
u  =  (/)  /\  ph ) ) } x  C_  y ) )
98rexeqbi1dv 2531 . . . . . 6  |-  ( 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 ) )
107, 9imbi12d 227 . . . . 5  |-  ( z  =  { u  e. 
{ (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) }  ->  ( ( E. w  w  e.  z  ->  E. x  e.  z  A. y  e.  z  x  C_  y
)  <->  ( 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 ) ) )
115, 10spcv 2663 . . . 4  |-  ( A. z ( E. w  w  e.  z  ->  E. x  e.  z  A. y  e.  z  x  C_  y )  ->  ( 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 ) )
123, 11ax-mp 7 . . 3  |-  ( 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 )
132, 12ax-mp 7 . 2  |-  E. x  e.  { u  e.  { (/)
,  { (/) } }  |  ( u  =  { (/) }  \/  (
u  =  (/)  /\  ph ) ) } A. y  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) } x  C_  y
141reg2exmidlema 4287 . 2  |-  ( E. x  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) } A. y  e.  { u  e.  { (/) ,  { (/) } }  |  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) } x  C_  y  ->  ( ph  \/  -.  ph ) )
1513, 14ax-mp 7 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    \/ wo 639   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   A.wral 2323   E.wrex 2324   {crab 2327    C_ wss 2945   (/)c0 3252   {csn 3403   {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410
This theorem is referenced by: (None)
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