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Theorem exmid1dc 4091
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 4085 or ordtriexmid 4405. 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 804 . . . . . . 7  |-  (DECID  x  =  { (/) }  ->  (
x  =  { (/) }  \/  -.  x  =  { (/) } ) )
31, 2syl 14 . . . . . 6  |-  ( (
ph  /\  x  C_  { (/) } )  ->  ( x  =  { (/) }  \/  -.  x  =  { (/) } ) )
4 df-ne 2284 . . . . . . . . 9  |-  ( x  =/=  { (/) }  <->  -.  x  =  { (/) } )
5 pwntru 4090 . . . . . . . . . 10  |-  ( ( x  C_  { (/) }  /\  x  =/=  { (/) } )  ->  x  =  (/) )
65ex 114 . . . . . . . . 9  |-  ( x 
C_  { (/) }  ->  ( x  =/=  { (/) }  ->  x  =  (/) ) )
74, 6syl5bir 152 . . . . . . . 8  |-  ( x 
C_  { (/) }  ->  ( -.  x  =  { (/)
}  ->  x  =  (/) ) )
87orim2d 760 . . . . . . 7  |-  ( x 
C_  { (/) }  ->  ( ( x  =  { (/)
}  \/  -.  x  =  { (/) } )  -> 
( x  =  { (/)
}  \/  x  =  (/) ) ) )
98adantl 273 . . . . . 6  |-  ( (
ph  /\  x  C_  { (/) } )  ->  ( (
x  =  { (/) }  \/  -.  x  =  { (/) } )  -> 
( x  =  { (/)
}  \/  x  =  (/) ) ) )
103, 9mpd 13 . . . . 5  |-  ( (
ph  /\  x  C_  { (/) } )  ->  ( x  =  { (/) }  \/  x  =  (/) ) )
1110orcomd 701 . . . 4  |-  ( (
ph  /\  x  C_  { (/) } )  ->  ( x  =  (/)  \/  x  =  { (/) } ) )
1211ex 114 . . 3  |-  ( ph  ->  ( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
1312alrimiv 1828 . 2  |-  ( ph  ->  A. x ( x 
C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
14 exmid01 4089 . 2  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
1513, 14sylibr 133 1  |-  ( ph  -> EXMID )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 680  DECID wdc 802   A.wal 1312    = wceq 1314    =/= wne 2283    C_ wss 3039   (/)c0 3331   {csn 3495  EXMIDwem 4086
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-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022
This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-exmid 4087
This theorem is referenced by:  exmidaclem  7028
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