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Theorem regexmid 4567
Description: The axiom of foundation implies excluded middle.

By foundation (or regularity), we mean the principle that every inhabited set has an element which is minimal (when arranged by  e.). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4569. (Contributed by Jim Kingdon, 3-Sep-2019.)

Hypothesis
Ref Expression
regexmid.1  |-  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )
Assertion
Ref Expression
regexmid  |-  ( ph  \/  -.  ph )
Distinct variable group:    ph, x, y, z

Proof of Theorem regexmid
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 eqid 2193 . . 3  |-  { w  e.  { (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  =  { w  e.  { (/) ,  { (/) } }  | 
( w  =  { (/)
}  \/  ( w  =  (/)  /\  ph )
) }
21regexmidlemm 4564 . 2  |-  E. y 
y  e.  { w  e.  { (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }
3 pp0ex 4218 . . . 4  |-  { (/) ,  { (/) } }  e.  _V
43rabex 4173 . . 3  |-  { w  e.  { (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  e.  _V
5 eleq2 2257 . . . . 5  |-  ( x  =  { w  e. 
{ (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  ( y  e.  x  <->  y  e.  { w  e.  { (/) ,  { (/) } }  | 
( w  =  { (/)
}  \/  ( w  =  (/)  /\  ph )
) } ) )
65exbidv 1836 . . . 4  |-  ( x  =  { w  e. 
{ (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  ( E. y  y  e.  x  <->  E. y  y  e. 
{ w  e.  { (/)
,  { (/) } }  |  ( w  =  { (/) }  \/  (
w  =  (/)  /\  ph ) ) } ) )
7 eleq2 2257 . . . . . . . . 9  |-  ( x  =  { w  e. 
{ (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  ( z  e.  x  <->  z  e.  { w  e.  { (/) ,  { (/) } }  | 
( w  =  { (/)
}  \/  ( w  =  (/)  /\  ph )
) } ) )
87notbid 668 . . . . . . . 8  |-  ( x  =  { w  e. 
{ (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  ( -.  z  e.  x  <->  -.  z  e.  { w  e.  { (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) } ) )
98imbi2d 230 . . . . . . 7  |-  ( x  =  { w  e. 
{ (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  ( ( z  e.  y  ->  -.  z  e.  x )  <->  ( z  e.  y  ->  -.  z  e.  { w  e.  { (/)
,  { (/) } }  |  ( w  =  { (/) }  \/  (
w  =  (/)  /\  ph ) ) } ) ) )
109albidv 1835 . . . . . 6  |-  ( x  =  { w  e. 
{ (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  ( A. z ( z  e.  y  ->  -.  z  e.  x )  <->  A. z ( z  e.  y  ->  -.  z  e.  { w  e.  { (/)
,  { (/) } }  |  ( w  =  { (/) }  \/  (
w  =  (/)  /\  ph ) ) } ) ) )
115, 10anbi12d 473 . . . . 5  |-  ( x  =  { w  e. 
{ (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  ( ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x )
)  <->  ( y  e. 
{ w  e.  { (/)
,  { (/) } }  |  ( w  =  { (/) }  \/  (
w  =  (/)  /\  ph ) ) }  /\  A. z ( z  e.  y  ->  -.  z  e.  { w  e.  { (/)
,  { (/) } }  |  ( w  =  { (/) }  \/  (
w  =  (/)  /\  ph ) ) } ) ) ) )
1211exbidv 1836 . . . 4  |-  ( x  =  { w  e. 
{ (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  ( E. y ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) )  <->  E. y
( y  e.  {
w  e.  { (/) ,  { (/) } }  | 
( w  =  { (/)
}  \/  ( w  =  (/)  /\  ph )
) }  /\  A. z ( z  e.  y  ->  -.  z  e.  { w  e.  { (/)
,  { (/) } }  |  ( w  =  { (/) }  \/  (
w  =  (/)  /\  ph ) ) } ) ) ) )
136, 12imbi12d 234 . . 3  |-  ( x  =  { w  e. 
{ (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  ( ( E. y  y  e.  x  ->  E. y
( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x )
) )  <->  ( E. y  y  e.  { w  e.  { (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  E. y ( y  e. 
{ w  e.  { (/)
,  { (/) } }  |  ( w  =  { (/) }  \/  (
w  =  (/)  /\  ph ) ) }  /\  A. z ( z  e.  y  ->  -.  z  e.  { w  e.  { (/)
,  { (/) } }  |  ( w  =  { (/) }  \/  (
w  =  (/)  /\  ph ) ) } ) ) ) ) )
14 regexmid.1 . . 3  |-  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )
154, 13, 14vtocl 2814 . 2  |-  ( E. y  y  e.  {
w  e.  { (/) ,  { (/) } }  | 
( w  =  { (/)
}  \/  ( w  =  (/)  /\  ph )
) }  ->  E. y
( y  e.  {
w  e.  { (/) ,  { (/) } }  | 
( w  =  { (/)
}  \/  ( w  =  (/)  /\  ph )
) }  /\  A. z ( z  e.  y  ->  -.  z  e.  { w  e.  { (/)
,  { (/) } }  |  ( w  =  { (/) }  \/  (
w  =  (/)  /\  ph ) ) } ) ) )
161regexmidlem1 4565 . 2  |-  ( E. y ( y  e. 
{ w  e.  { (/)
,  { (/) } }  |  ( w  =  { (/) }  \/  (
w  =  (/)  /\  ph ) ) }  /\  A. z ( z  e.  y  ->  -.  z  e.  { w  e.  { (/)
,  { (/) } }  |  ( w  =  { (/) }  \/  (
w  =  (/)  /\  ph ) ) } ) )  ->  ( 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   A.wal 1362    = wceq 1364   E.wex 1503    e. wcel 2164   {crab 2476   (/)c0 3446   {csn 3618   {cpr 3619
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 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625
This theorem is referenced by: (None)
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