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Theorem undifexmid 4085
Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3411 and undifdcss 6777 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 4023 . . . . 5  |-  (/)  e.  _V
21snid 3524 . . . 4  |-  (/)  e.  { (/)
}
3 ssrab2 3150 . . . . 5  |-  { z  e.  { (/) }  |  ph }  C_  { (/) }
4 p0ex 4080 . . . . . . 7  |-  { (/) }  e.  _V
54rabex 4040 . . . . . 6  |-  { z  e.  { (/) }  |  ph }  e.  _V
6 sseq12 3090 . . . . . . 7  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( x  C_  y  <->  { z  e.  { (/) }  |  ph }  C_  {
(/) } ) )
7 simpl 108 . . . . . . . . 9  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  ->  x  =  { z  e.  { (/) }  |  ph } )
8 simpr 109 . . . . . . . . . 10  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
y  =  { (/) } )
98, 7difeq12d 3163 . . . . . . . . 9  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( y  \  x
)  =  ( {
(/) }  \  { z  e.  { (/) }  |  ph } ) )
107, 9uneq12d 3199 . . . . . . . 8  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( x  u.  (
y  \  x )
)  =  ( { z  e.  { (/) }  |  ph }  u.  ( { (/) }  \  {
z  e.  { (/) }  |  ph } ) ) )
1110, 8eqeq12d 2130 . . . . . . 7  |-  ( ( x  =  { z  e.  { (/) }  |  ph }  /\  y  =  { (/) } )  -> 
( ( x  u.  ( y  \  x
) )  =  y  <-> 
( { z  e. 
{ (/) }  |  ph }  u.  ( { (/)
}  \  { z  e.  { (/) }  |  ph } ) )  =  { (/) } ) )
126, 11bibi12d 234 . . . . . 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 2713 . . . . 5  |-  ( { z  e.  { (/) }  |  ph }  C_  {
(/) }  <->  ( { z  e.  { (/) }  |  ph }  u.  ( {
(/) }  \  { z  e.  { (/) }  |  ph } ) )  =  { (/) } )
153, 14mpbi 144 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  u.  ( { (/) }  \  {
z  e.  { (/) }  |  ph } ) )  =  { (/) }
162, 15eleqtrri 2191 . . 3  |-  (/)  e.  ( { z  e.  { (/)
}  |  ph }  u.  ( { (/) }  \  { z  e.  { (/)
}  |  ph }
) )
17 elun 3185 . . 3  |-  ( (/)  e.  ( { z  e. 
{ (/) }  |  ph }  u.  ( { (/)
}  \  { z  e.  { (/) }  |  ph } ) )  <->  ( (/)  e.  {
z  e.  { (/) }  |  ph }  \/  (/) 
e.  ( { (/) } 
\  { z  e. 
{ (/) }  |  ph } ) ) )
1816, 17mpbi 144 . 2  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  \/  (/)  e.  ( {
(/) }  \  { z  e.  { (/) }  |  ph } ) )
19 biidd 171 . . . . . 6  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
2019elrab3 2812 . . . . 5  |-  ( (/)  e.  { (/) }  ->  ( (/) 
e.  { z  e. 
{ (/) }  |  ph } 
<-> 
ph ) )
212, 20ax-mp 5 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  ph )
2221biimpi 119 . . 3  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  ->  ph )
23 eldifn 3167 . . . 4  |-  ( (/)  e.  ( { (/) }  \  { z  e.  { (/)
}  |  ph }
)  ->  -.  (/)  e.  {
z  e.  { (/) }  |  ph } )
2423, 21sylnib 648 . . 3  |-  ( (/)  e.  ( { (/) }  \  { z  e.  { (/)
}  |  ph }
)  ->  -.  ph )
2522, 24orim12i 731 . 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 103    <-> wb 104    \/ wo 680    = wceq 1314    e. wcel 1463   {crab 2395    \ cdif 3036    u. cun 3037    C_ wss 3039   (/)c0 3331   {csn 3495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501
This theorem is referenced by: (None)
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