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Theorem undifexmid 4208
Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3518 and undifdcss 6941 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)
Hypothesis
Ref Expression
undifexmid.1  |-  ( x 
C_  y  <->  ( x  u.  ( y  \  x
) )  =  y )
Assertion
Ref Expression
undifexmid  |-  ( ph  \/  -.  ph )
Distinct variable group:    ph, x, y

Proof of Theorem undifexmid
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 0ex 4145 . . . . 5  |-  (/)  e.  _V
21snid 3638 . . . 4  |-  (/)  e.  { (/)
}
3 ssrab2 3255 . . . . 5  |-  { z  e.  { (/) }  |  ph }  C_  { (/) }
4 p0ex 4203 . . . . . . 7  |-  { (/) }  e.  _V
54rabex 4162 . . . . . 6  |-  { z  e.  { (/) }  |  ph }  e.  _V
6 sseq12 3195 . . . . . . 7  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( x  C_  y  <->  { z  e.  { (/) }  |  ph }  C_  {
(/) } ) )
7 simpl 109 . . . . . . . . 9  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  ->  x  =  { z  e.  { (/) }  |  ph } )
8 simpr 110 . . . . . . . . . 10  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
y  =  { (/) } )
98, 7difeq12d 3269 . . . . . . . . 9  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( y  \  x
)  =  ( {
(/) }  \  { z  e.  { (/) }  |  ph } ) )
107, 9uneq12d 3305 . . . . . . . 8  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( x  u.  (
y  \  x )
)  =  ( { z  e.  { (/) }  |  ph }  u.  ( { (/) }  \  {
z  e.  { (/) }  |  ph } ) ) )
1110, 8eqeq12d 2204 . . . . . . 7  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( ( x  u.  ( y  \  x
) )  =  y  <-> 
( { z  e. 
{ (/) }  |  ph }  u.  ( { (/)
}  \  { z  e.  { (/) }  |  ph } ) )  =  { (/) } ) )
126, 11bibi12d 235 . . . . . 6  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( ( x  C_  y 
<->  ( x  u.  (
y  \  x )
)  =  y )  <-> 
( { z  e. 
{ (/) }  |  ph }  C_  { (/) }  <->  ( {
z  e.  { (/) }  |  ph }  u.  ( { (/) }  \  {
z  e.  { (/) }  |  ph } ) )  =  { (/) } ) ) )
13 undifexmid.1 . . . . . 6  |-  ( x 
C_  y  <->  ( x  u.  ( y  \  x
) )  =  y )
145, 4, 12, 13vtocl2 2807 . . . . 5  |-  ( { z  e.  { (/) }  |  ph }  C_  {
(/) }  <->  ( { z  e.  { (/) }  |  ph }  u.  ( {
(/) }  \  { z  e.  { (/) }  |  ph } ) )  =  { (/) } )
153, 14mpbi 145 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  u.  ( { (/) }  \  {
z  e.  { (/) }  |  ph } ) )  =  { (/) }
162, 15eleqtrri 2265 . . 3  |-  (/)  e.  ( { z  e.  { (/)
}  |  ph }  u.  ( { (/) }  \  { z  e.  { (/)
}  |  ph }
) )
17 elun 3291 . . 3  |-  ( (/)  e.  ( { z  e. 
{ (/) }  |  ph }  u.  ( { (/)
}  \  { z  e.  { (/) }  |  ph } ) )  <->  ( (/)  e.  {
z  e.  { (/) }  |  ph }  \/  (/) 
e.  ( { (/) } 
\  { z  e. 
{ (/) }  |  ph } ) ) )
1816, 17mpbi 145 . 2  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  \/  (/)  e.  ( {
(/) }  \  { z  e.  { (/) }  |  ph } ) )
19 biidd 172 . . . . . 6  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
2019elrab3 2909 . . . . 5  |-  ( (/)  e.  { (/) }  ->  ( (/) 
e.  { z  e. 
{ (/) }  |  ph } 
<-> 
ph ) )
212, 20ax-mp 5 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  ph )
2221biimpi 120 . . 3  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  ->  ph )
23 eldifn 3273 . . . 4  |-  ( (/)  e.  ( { (/) }  \  { z  e.  { (/)
}  |  ph }
)  ->  -.  (/)  e.  {
z  e.  { (/) }  |  ph } )
2423, 21sylnib 677 . . 3  |-  ( (/)  e.  ( { (/) }  \  { z  e.  { (/)
}  |  ph }
)  ->  -.  ph )
2522, 24orim12i 760 . 2  |-  ( (
(/)  e.  { z  e.  { (/) }  |  ph }  \/  (/)  e.  ( { (/) }  \  {
z  e.  { (/) }  |  ph } ) )  ->  ( ph  \/  -.  ph ) )
2618, 25ax-mp 5 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160   {crab 2472    \ cdif 3141    u. cun 3142    C_ wss 3144   (/)c0 3437   {csn 3607
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 4189
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  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
This theorem is referenced by: (None)
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