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Theorem undifexmid 4195
Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3505 and undifdcss 6925 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 4132 . . . . 5  |-  (/)  e.  _V
21snid 3625 . . . 4  |-  (/)  e.  { (/)
}
3 ssrab2 3242 . . . . 5  |-  { z  e.  { (/) }  |  ph }  C_  { (/) }
4 p0ex 4190 . . . . . . 7  |-  { (/) }  e.  _V
54rabex 4149 . . . . . 6  |-  { z  e.  { (/) }  |  ph }  e.  _V
6 sseq12 3182 . . . . . . 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 3256 . . . . . . . . 9  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( y  \  x
)  =  ( {
(/) }  \  { z  e.  { (/) }  |  ph } ) )
107, 9uneq12d 3292 . . . . . . . 8  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( x  u.  (
y  \  x )
)  =  ( { z  e.  { (/) }  |  ph }  u.  ( { (/) }  \  {
z  e.  { (/) }  |  ph } ) ) )
1110, 8eqeq12d 2192 . . . . . . 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 2794 . . . . 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 2253 . . 3  |-  (/)  e.  ( { z  e.  { (/)
}  |  ph }  u.  ( { (/) }  \  { z  e.  { (/)
}  |  ph }
) )
17 elun 3278 . . 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 2896 . . . . 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 3260 . . . 4  |-  ( (/)  e.  ( { (/) }  \  { z  e.  { (/)
}  |  ph }
)  ->  -.  (/)  e.  {
z  e.  { (/) }  |  ph } )
2423, 21sylnib 676 . . 3  |-  ( (/)  e.  ( { (/) }  \  { z  e.  { (/)
}  |  ph }
)  ->  -.  ph )
2522, 24orim12i 759 . 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 708    = wceq 1353    e. wcel 2148   {crab 2459    \ cdif 3128    u. cun 3129    C_ wss 3131   (/)c0 3424   {csn 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 4123  ax-nul 4131  ax-pow 4176
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-ral 2460  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600
This theorem is referenced by: (None)
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