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Theorem exmidaclem 7028
Description: Lemma for exmidac 7029. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.)
Hypotheses
Ref Expression
exmidaclem.a  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  y  =  { (/)
} ) }
exmidaclem.b  |-  B  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  y  =  { (/) } ) }
exmidaclem.c  |-  C  =  { A ,  B }
Assertion
Ref Expression
exmidaclem  |-  (CHOICE  -> EXMID )
Distinct variable groups:    x, A    x, B    x, y
Allowed substitution hints:    A( y)    B( y)    C( x, y)

Proof of Theorem exmidaclem
Dummy variables  z  f  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 108 . . . 4  |-  ( (CHOICE  /\  y  C_  { (/) } )  -> CHOICE
)
2 exmidaclem.c . . . . . 6  |-  C  =  { A ,  B }
3 exmidaclem.a . . . . . . . 8  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  y  =  { (/)
} ) }
4 pp0ex 4081 . . . . . . . . 9  |-  { (/) ,  { (/) } }  e.  _V
54rabex 4040 . . . . . . . 8  |-  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  y  =  { (/) } ) }  e.  _V
63, 5eqeltri 2188 . . . . . . 7  |-  A  e. 
_V
7 exmidaclem.b . . . . . . . 8  |-  B  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  y  =  { (/) } ) }
84rabex 4040 . . . . . . . 8  |-  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/) }  \/  y  =  { (/) } ) }  e.  _V
97, 8eqeltri 2188 . . . . . . 7  |-  B  e. 
_V
10 prexg 4101 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
116, 9, 10mp2an 420 . . . . . 6  |-  { A ,  B }  e.  _V
122, 11eqeltri 2188 . . . . 5  |-  C  e. 
_V
1312a1i 9 . . . 4  |-  ( (CHOICE  /\  y  C_  { (/) } )  ->  C  e.  _V )
14 simpr 109 . . . . . . 7  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  z  e.  C )  ->  z  e.  C )
1514, 2syl6eleq 2208 . . . . . 6  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  z  e.  C )  ->  z  e.  { A ,  B } )
16 elpri 3518 . . . . . 6  |-  ( z  e.  { A ,  B }  ->  ( z  =  A  \/  z  =  B ) )
17 0ex 4023 . . . . . . . . . . 11  |-  (/)  e.  _V
1817prid1 3597 . . . . . . . . . 10  |-  (/)  e.  { (/)
,  { (/) } }
19 eqid 2115 . . . . . . . . . . 11  |-  (/)  =  (/)
2019orci 703 . . . . . . . . . 10  |-  ( (/)  =  (/)  \/  y  =  { (/) } )
21 eqeq1 2122 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
2221orbi1d 763 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( ( x  =  (/)  \/  y  =  { (/) } )  <->  ( (/)  =  (/)  \/  y  =  { (/) } ) ) )
2322, 3elrab2 2814 . . . . . . . . . 10  |-  ( (/)  e.  A  <->  ( (/)  e.  { (/)
,  { (/) } }  /\  ( (/)  =  (/)  \/  y  =  { (/) } ) ) )
2418, 20, 23mpbir2an 909 . . . . . . . . 9  |-  (/)  e.  A
25 eleq2 2179 . . . . . . . . 9  |-  ( z  =  A  ->  ( (/) 
e.  z  <->  (/)  e.  A
) )
2624, 25mpbiri 167 . . . . . . . 8  |-  ( z  =  A  ->  (/)  e.  z )
27 elex2 2674 . . . . . . . 8  |-  ( (/)  e.  z  ->  E. w  w  e.  z )
2826, 27syl 14 . . . . . . 7  |-  ( z  =  A  ->  E. w  w  e.  z )
29 p0ex 4080 . . . . . . . . . . 11  |-  { (/) }  e.  _V
3029prid2 3598 . . . . . . . . . 10  |-  { (/) }  e.  { (/) ,  { (/)
} }
31 eqid 2115 . . . . . . . . . . 11  |-  { (/) }  =  { (/) }
3231orci 703 . . . . . . . . . 10  |-  ( {
(/) }  =  { (/)
}  \/  y  =  { (/) } )
33 eqeq1 2122 . . . . . . . . . . . 12  |-  ( x  =  { (/) }  ->  ( x  =  { (/) }  <->  { (/) }  =  { (/)
} ) )
3433orbi1d 763 . . . . . . . . . . 11  |-  ( x  =  { (/) }  ->  ( ( x  =  { (/)
}  \/  y  =  { (/) } )  <->  ( { (/)
}  =  { (/) }  \/  y  =  { (/)
} ) ) )
3534, 7elrab2 2814 . . . . . . . . . 10  |-  ( {
(/) }  e.  B  <->  ( { (/) }  e.  { (/)
,  { (/) } }  /\  ( { (/) }  =  { (/) }  \/  y  =  { (/) } ) ) )
3630, 32, 35mpbir2an 909 . . . . . . . . 9  |-  { (/) }  e.  B
37 eleq2 2179 . . . . . . . . 9  |-  ( z  =  B  ->  ( { (/) }  e.  z  <->  { (/) }  e.  B
) )
3836, 37mpbiri 167 . . . . . . . 8  |-  ( z  =  B  ->  { (/) }  e.  z )
39 elex2 2674 . . . . . . . 8  |-  ( {
(/) }  e.  z  ->  E. w  w  e.  z )
4038, 39syl 14 . . . . . . 7  |-  ( z  =  B  ->  E. w  w  e.  z )
4128, 40jaoi 688 . . . . . 6  |-  ( ( z  =  A  \/  z  =  B )  ->  E. w  w  e.  z )
4215, 16, 413syl 17 . . . . 5  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  z  e.  C )  ->  E. w  w  e.  z )
4342ralrimiva 2480 . . . 4  |-  ( (CHOICE  /\  y  C_  { (/) } )  ->  A. z  e.  C  E. w  w  e.  z )
441, 13, 43acfun 7027 . . 3  |-  ( (CHOICE  /\  y  C_  { (/) } )  ->  E. f ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )
45 0nep0 4057 . . . . . . . . . 10  |-  (/)  =/=  { (/)
}
4645neii 2285 . . . . . . . . 9  |-  -.  (/)  =  { (/)
}
47 simplr 502 . . . . . . . . . 10  |-  ( ( ( ( (CHOICE  /\  y  C_ 
{ (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  ( f `  z
)  e.  z ) )  /\  ( f `
 A )  =  (/) )  /\  (
f `  B )  =  { (/) } )  -> 
( f `  A
)  =  (/) )
48 simpr 109 . . . . . . . . . 10  |-  ( ( ( ( (CHOICE  /\  y  C_ 
{ (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  ( f `  z
)  e.  z ) )  /\  ( f `
 A )  =  (/) )  /\  (
f `  B )  =  { (/) } )  -> 
( f `  B
)  =  { (/) } )
4947, 48eqeq12d 2130 . . . . . . . . 9  |-  ( ( ( ( (CHOICE  /\  y  C_ 
{ (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  ( f `  z
)  e.  z ) )  /\  ( f `
 A )  =  (/) )  /\  (
f `  B )  =  { (/) } )  -> 
( ( f `  A )  =  ( f `  B )  <->  (/)  =  { (/) } ) )
5046, 49mtbiri 647 . . . . . . . 8  |-  ( ( ( ( (CHOICE  /\  y  C_ 
{ (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  ( f `  z
)  e.  z ) )  /\  ( f `
 A )  =  (/) )  /\  (
f `  B )  =  { (/) } )  ->  -.  ( f `  A
)  =  ( f `
 B ) )
51 olc 683 . . . . . . . . . . . . 13  |-  ( y  =  { (/) }  ->  ( x  =  (/)  \/  y  =  { (/) } ) )
5251ralrimivw 2481 . . . . . . . . . . . 12  |-  ( y  =  { (/) }  ->  A. x  e.  { (/) ,  { (/) } }  (
x  =  (/)  \/  y  =  { (/) } ) )
53 rabid2 2582 . . . . . . . . . . . 12  |-  ( {
(/) ,  { (/) } }  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  y  =  { (/)
} ) }  <->  A. x  e.  { (/) ,  { (/) } }  ( x  =  (/)  \/  y  =  { (/)
} ) )
5452, 53sylibr 133 . . . . . . . . . . 11  |-  ( y  =  { (/) }  ->  {
(/) ,  { (/) } }  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  y  =  { (/)
} ) } )
5554, 3syl6eqr 2166 . . . . . . . . . 10  |-  ( y  =  { (/) }  ->  {
(/) ,  { (/) } }  =  A )
56 olc 683 . . . . . . . . . . . . 13  |-  ( y  =  { (/) }  ->  ( x  =  { (/) }  \/  y  =  { (/)
} ) )
5756ralrimivw 2481 . . . . . . . . . . . 12  |-  ( y  =  { (/) }  ->  A. x  e.  { (/) ,  { (/) } }  (
x  =  { (/) }  \/  y  =  { (/)
} ) )
58 rabid2 2582 . . . . . . . . . . . 12  |-  ( {
(/) ,  { (/) } }  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  y  =  { (/) } ) }  <->  A. x  e.  { (/) ,  { (/) } }  (
x  =  { (/) }  \/  y  =  { (/)
} ) )
5957, 58sylibr 133 . . . . . . . . . . 11  |-  ( y  =  { (/) }  ->  {
(/) ,  { (/) } }  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  y  =  { (/) } ) } )
6059, 7syl6eqr 2166 . . . . . . . . . 10  |-  ( y  =  { (/) }  ->  {
(/) ,  { (/) } }  =  B )
6155, 60eqtr3d 2150 . . . . . . . . 9  |-  ( y  =  { (/) }  ->  A  =  B )
6261fveq2d 5391 . . . . . . . 8  |-  ( y  =  { (/) }  ->  ( f `  A )  =  ( f `  B ) )
6350, 62nsyl 600 . . . . . . 7  |-  ( ( ( ( (CHOICE  /\  y  C_ 
{ (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  ( f `  z
)  e.  z ) )  /\  ( f `
 A )  =  (/) )  /\  (
f `  B )  =  { (/) } )  ->  -.  y  =  { (/)
} )
6463olcd 706 . . . . . 6  |-  ( ( ( ( (CHOICE  /\  y  C_ 
{ (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  ( f `  z
)  e.  z ) )  /\  ( f `
 A )  =  (/) )  /\  (
f `  B )  =  { (/) } )  -> 
( y  =  { (/)
}  \/  -.  y  =  { (/) } ) )
65 simpr 109 . . . . . . 7  |-  ( ( ( ( (CHOICE  /\  y  C_ 
{ (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  ( f `  z
)  e.  z ) )  /\  ( f `
 A )  =  (/) )  /\  y  =  { (/) } )  -> 
y  =  { (/) } )
6665orcd 705 . . . . . 6  |-  ( ( ( ( (CHOICE  /\  y  C_ 
{ (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  ( f `  z
)  e.  z ) )  /\  ( f `
 A )  =  (/) )  /\  y  =  { (/) } )  -> 
( y  =  { (/)
}  \/  -.  y  =  { (/) } ) )
67 fveq2 5387 . . . . . . . . . . 11  |-  ( z  =  B  ->  (
f `  z )  =  ( f `  B ) )
68 id 19 . . . . . . . . . . 11  |-  ( z  =  B  ->  z  =  B )
6967, 68eleq12d 2186 . . . . . . . . . 10  |-  ( z  =  B  ->  (
( f `  z
)  e.  z  <->  ( f `  B )  e.  B
) )
70 simprr 504 . . . . . . . . . 10  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  ->  A. z  e.  C  ( f `  z
)  e.  z )
719prid2 3598 . . . . . . . . . . . 12  |-  B  e. 
{ A ,  B }
7271, 2eleqtrri 2191 . . . . . . . . . . 11  |-  B  e.  C
7372a1i 9 . . . . . . . . . 10  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  ->  B  e.  C )
7469, 70, 73rspcdva 2766 . . . . . . . . 9  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  -> 
( f `  B
)  e.  B )
75 eqeq1 2122 . . . . . . . . . . 11  |-  ( x  =  ( f `  B )  ->  (
x  =  { (/) }  <-> 
( f `  B
)  =  { (/) } ) )
7675orbi1d 763 . . . . . . . . . 10  |-  ( x  =  ( f `  B )  ->  (
( x  =  { (/)
}  \/  y  =  { (/) } )  <->  ( (
f `  B )  =  { (/) }  \/  y  =  { (/) } ) ) )
7776, 7elrab2 2814 . . . . . . . . 9  |-  ( ( f `  B )  e.  B  <->  ( (
f `  B )  e.  { (/) ,  { (/) } }  /\  ( ( f `  B )  =  { (/) }  \/  y  =  { (/) } ) ) )
7874, 77sylib 121 . . . . . . . 8  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  -> 
( ( f `  B )  e.  { (/)
,  { (/) } }  /\  ( ( f `  B )  =  { (/)
}  \/  y  =  { (/) } ) ) )
7978simprd 113 . . . . . . 7  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  -> 
( ( f `  B )  =  { (/)
}  \/  y  =  { (/) } ) )
8079adantr 272 . . . . . 6  |-  ( ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  /\  ( f `  A
)  =  (/) )  -> 
( ( f `  B )  =  { (/)
}  \/  y  =  { (/) } ) )
8164, 66, 80mpjaodan 770 . . . . 5  |-  ( ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  /\  ( f `  A
)  =  (/) )  -> 
( y  =  { (/)
}  \/  -.  y  =  { (/) } ) )
82 df-dc 803 . . . . 5  |-  (DECID  y  =  { (/) }  <->  ( y  =  { (/) }  \/  -.  y  =  { (/) } ) )
8381, 82sylibr 133 . . . 4  |-  ( ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  /\  ( f `  A
)  =  (/) )  -> DECID  y  =  { (/) } )
84 simpr 109 . . . . . 6  |-  ( ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  /\  y  =  { (/) } )  ->  y  =  { (/)
} )
8584orcd 705 . . . . 5  |-  ( ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  /\  y  =  { (/) } )  ->  ( y  =  { (/) }  \/  -.  y  =  { (/) } ) )
8685, 82sylibr 133 . . . 4  |-  ( ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  /\  y  =  { (/) } )  -> DECID 
y  =  { (/) } )
87 fveq2 5387 . . . . . . . 8  |-  ( z  =  A  ->  (
f `  z )  =  ( f `  A ) )
88 id 19 . . . . . . . 8  |-  ( z  =  A  ->  z  =  A )
8987, 88eleq12d 2186 . . . . . . 7  |-  ( z  =  A  ->  (
( f `  z
)  e.  z  <->  ( f `  A )  e.  A
) )
906prid1 3597 . . . . . . . . 9  |-  A  e. 
{ A ,  B }
9190, 2eleqtrri 2191 . . . . . . . 8  |-  A  e.  C
9291a1i 9 . . . . . . 7  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  ->  A  e.  C )
9389, 70, 92rspcdva 2766 . . . . . 6  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  -> 
( f `  A
)  e.  A )
94 eqeq1 2122 . . . . . . . 8  |-  ( x  =  ( f `  A )  ->  (
x  =  (/)  <->  ( f `  A )  =  (/) ) )
9594orbi1d 763 . . . . . . 7  |-  ( x  =  ( f `  A )  ->  (
( x  =  (/)  \/  y  =  { (/) } )  <->  ( ( f `
 A )  =  (/)  \/  y  =  { (/)
} ) ) )
9695, 3elrab2 2814 . . . . . 6  |-  ( ( f `  A )  e.  A  <->  ( (
f `  A )  e.  { (/) ,  { (/) } }  /\  ( ( f `  A )  =  (/)  \/  y  =  { (/) } ) ) )
9793, 96sylib 121 . . . . 5  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  -> 
( ( f `  A )  e.  { (/)
,  { (/) } }  /\  ( ( f `  A )  =  (/)  \/  y  =  { (/) } ) ) )
9897simprd 113 . . . 4  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  -> 
( ( f `  A )  =  (/)  \/  y  =  { (/) } ) )
9983, 86, 98mpjaodan 770 . . 3  |-  ( ( (CHOICE 
/\  y  C_  { (/) } )  /\  ( f  Fn  C  /\  A. z  e.  C  (
f `  z )  e.  z ) )  -> DECID  y  =  { (/) } )
10044, 99exlimddv 1852 . 2  |-  ( (CHOICE  /\  y  C_  { (/) } )  -> DECID 
y  =  { (/) } )
101100exmid1dc 4091 1  |-  (CHOICE  -> EXMID )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 680  DECID wdc 802    = wceq 1314   E.wex 1451    e. wcel 1463   A.wral 2391   {crab 2395   _Vcvv 2658    C_ wss 3039   (/)c0 3331   {csn 3495   {cpr 3496  EXMIDwem 4086    Fn wfn 5086   ` cfv 5091  CHOICEwac 7025
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-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-exmid 4087  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ac 7026
This theorem is referenced by:  exmidac  7029
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