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Theorem undifexmid 4205
Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3515 and undifdcss 6935 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 4142 . . . . 5  |-  (/)  e.  _V
21snid 3635 . . . 4  |-  (/)  e.  { (/)
}
3 ssrab2 3252 . . . . 5  |-  { z  e.  { (/) }  |  ph }  C_  { (/) }
4 p0ex 4200 . . . . . . 7  |-  { (/) }  e.  _V
54rabex 4159 . . . . . 6  |-  { z  e.  { (/) }  |  ph }  e.  _V
6 sseq12 3192 . . . . . . 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 3266 . . . . . . . . 9  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( y  \  x
)  =  ( {
(/) }  \  { z  e.  { (/) }  |  ph } ) )
107, 9uneq12d 3302 . . . . . . . 8  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( x  u.  (
y  \  x )
)  =  ( { z  e.  { (/) }  |  ph }  u.  ( { (/) }  \  {
z  e.  { (/) }  |  ph } ) ) )
1110, 8eqeq12d 2202 . . . . . . 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 2804 . . . . 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 2263 . . 3  |-  (/)  e.  ( { z  e.  { (/)
}  |  ph }  u.  ( { (/) }  \  { z  e.  { (/)
}  |  ph }
) )
17 elun 3288 . . 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 2906 . . . . 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 3270 . . . 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 1363    e. wcel 2158   {crab 2469    \ cdif 3138    u. cun 3139    C_ wss 3141   (/)c0 3434   {csn 3604
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610
This theorem is referenced by: (None)
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