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Theorem regexmid 4535
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 4537. (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 2177 . . 3  |-  { w  e.  { (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  =  { w  e.  { (/) ,  { (/) } }  | 
( w  =  { (/)
}  \/  ( w  =  (/)  /\  ph )
) }
21regexmidlemm 4532 . 2  |-  E. y 
y  e.  { w  e.  { (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }
3 pp0ex 4190 . . . 4  |-  { (/) ,  { (/) } }  e.  _V
43rabex 4148 . . 3  |-  { w  e.  { (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  e.  _V
5 eleq2 2241 . . . . 5  |-  ( x  =  { w  e. 
{ (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  ( y  e.  x  <->  y  e.  { w  e.  { (/) ,  { (/) } }  | 
( w  =  { (/)
}  \/  ( w  =  (/)  /\  ph )
) } ) )
65exbidv 1825 . . . 4  |-  ( x  =  { w  e. 
{ (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  ( E. y  y  e.  x  <->  E. y  y  e. 
{ w  e.  { (/)
,  { (/) } }  |  ( w  =  { (/) }  \/  (
w  =  (/)  /\  ph ) ) } ) )
7 eleq2 2241 . . . . . . . . 9  |-  ( x  =  { w  e. 
{ (/) ,  { (/) } }  |  ( w  =  { (/) }  \/  ( w  =  (/)  /\  ph ) ) }  ->  ( z  e.  x  <->  z  e.  { w  e.  { (/) ,  { (/) } }  | 
( w  =  { (/)
}  \/  ( w  =  (/)  /\  ph )
) } ) )
87notbid 667 . . . . . . . 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 1824 . . . . . 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 1825 . . . 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 2792 . 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 4533 . 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 708   A.wal 1351    = wceq 1353   E.wex 1492    e. wcel 2148   {crab 2459   (/)c0 3423   {csn 3593   {cpr 3594
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600
This theorem is referenced by: (None)
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