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Theorem exmid1dc 4197
Description: A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4190 or ordtriexmid 4516. In this context  x  =  { (/) } can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
Hypothesis
Ref Expression
exmid1dc.x  |-  ( (
ph  /\  x  C_  { (/) } )  -> DECID  x  =  { (/)
} )
Assertion
Ref Expression
exmid1dc  |-  ( ph  -> EXMID )
Distinct variable group:    ph, x

Proof of Theorem exmid1dc
StepHypRef Expression
1 exmid1dc.x . . . . . . 7  |-  ( (
ph  /\  x  C_  { (/) } )  -> DECID  x  =  { (/)
} )
2 exmiddc 836 . . . . . . 7  |-  (DECID  x  =  { (/) }  ->  (
x  =  { (/) }  \/  -.  x  =  { (/) } ) )
31, 2syl 14 . . . . . 6  |-  ( (
ph  /\  x  C_  { (/) } )  ->  ( x  =  { (/) }  \/  -.  x  =  { (/) } ) )
4 df-ne 2348 . . . . . . . . 9  |-  ( x  =/=  { (/) }  <->  -.  x  =  { (/) } )
5 pwntru 4196 . . . . . . . . . 10  |-  ( ( x  C_  { (/) }  /\  x  =/=  { (/) } )  ->  x  =  (/) )
65ex 115 . . . . . . . . 9  |-  ( x 
C_  { (/) }  ->  ( x  =/=  { (/) }  ->  x  =  (/) ) )
74, 6biimtrrid 153 . . . . . . . 8  |-  ( x 
C_  { (/) }  ->  ( -.  x  =  { (/)
}  ->  x  =  (/) ) )
87orim2d 788 . . . . . . 7  |-  ( x 
C_  { (/) }  ->  ( ( x  =  { (/)
}  \/  -.  x  =  { (/) } )  -> 
( x  =  { (/)
}  \/  x  =  (/) ) ) )
98adantl 277 . . . . . 6  |-  ( (
ph  /\  x  C_  { (/) } )  ->  ( (
x  =  { (/) }  \/  -.  x  =  { (/) } )  -> 
( x  =  { (/)
}  \/  x  =  (/) ) ) )
103, 9mpd 13 . . . . 5  |-  ( (
ph  /\  x  C_  { (/) } )  ->  ( x  =  { (/) }  \/  x  =  (/) ) )
1110orcomd 729 . . . 4  |-  ( (
ph  /\  x  C_  { (/) } )  ->  ( x  =  (/)  \/  x  =  { (/) } ) )
1211ex 115 . . 3  |-  ( ph  ->  ( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
1312alrimiv 1874 . 2  |-  ( ph  ->  A. x ( x 
C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
14 exmid01 4195 . 2  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
1513, 14sylibr 134 1  |-  ( ph  -> EXMID )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 708  DECID wdc 834   A.wal 1351    = wceq 1353    =/= wne 2347    C_ wss 3129   (/)c0 3422   {csn 3591  EXMIDwem 4191
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-ext 2159  ax-nul 4126
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3597  df-exmid 4192
This theorem is referenced by:  pw1fin  6903  exmidonfin  7186  exmidaclem  7200  exmidontri  7231  exmidontri2or  7235
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