ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  regexmidlem1 Unicode version

Theorem regexmidlem1 4546
Description: Lemma for regexmid 4548. If  A has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.)
Hypothesis
Ref Expression
regexmidlemm.a  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
Assertion
Ref Expression
regexmidlem1  |-  ( E. y ( y  e.  A  /\  A. z
( z  e.  y  ->  -.  z  e.  A ) )  -> 
( ph  \/  -.  ph ) )
Distinct variable groups:    y, A, z    ph, x, y
Allowed substitution hints:    ph( z)    A( x)

Proof of Theorem regexmidlem1
StepHypRef Expression
1 eqeq1 2195 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  { (/) }  <-> 
y  =  { (/) } ) )
2 eqeq1 2195 . . . . . . . 8  |-  ( x  =  y  ->  (
x  =  (/)  <->  y  =  (/) ) )
32anbi1d 465 . . . . . . 7  |-  ( x  =  y  ->  (
( x  =  (/)  /\ 
ph )  <->  ( y  =  (/)  /\  ph )
) )
41, 3orbi12d 794 . . . . . 6  |-  ( x  =  y  ->  (
( x  =  { (/)
}  \/  ( x  =  (/)  /\  ph )
)  <->  ( y  =  { (/) }  \/  (
y  =  (/)  /\  ph ) ) ) )
5 regexmidlemm.a . . . . . 6  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
64, 5elrab2 2910 . . . . 5  |-  ( y  e.  A  <->  ( y  e.  { (/) ,  { (/) } }  /\  ( y  =  { (/) }  \/  ( y  =  (/)  /\ 
ph ) ) ) )
76simprbi 275 . . . 4  |-  ( y  e.  A  ->  (
y  =  { (/) }  \/  ( y  =  (/)  /\  ph ) ) )
8 0ex 4144 . . . . . . . . 9  |-  (/)  e.  _V
98snid 3637 . . . . . . . 8  |-  (/)  e.  { (/)
}
10 eleq2 2252 . . . . . . . 8  |-  ( y  =  { (/) }  ->  (
(/)  e.  y  <->  (/)  e.  { (/)
} ) )
119, 10mpbiri 168 . . . . . . 7  |-  ( y  =  { (/) }  ->  (/)  e.  y )
12 eleq1 2251 . . . . . . . . 9  |-  ( z  =  (/)  ->  ( z  e.  y  <->  (/)  e.  y ) )
13 eleq1 2251 . . . . . . . . . 10  |-  ( z  =  (/)  ->  ( z  e.  A  <->  (/)  e.  A
) )
1413notbid 668 . . . . . . . . 9  |-  ( z  =  (/)  ->  ( -.  z  e.  A  <->  -.  (/)  e.  A
) )
1512, 14imbi12d 234 . . . . . . . 8  |-  ( z  =  (/)  ->  ( ( z  e.  y  ->  -.  z  e.  A
)  <->  ( (/)  e.  y  ->  -.  (/)  e.  A
) ) )
168, 15spcv 2845 . . . . . . 7  |-  ( A. z ( z  e.  y  ->  -.  z  e.  A )  ->  ( (/) 
e.  y  ->  -.  (/) 
e.  A ) )
1711, 16syl5com 29 . . . . . 6  |-  ( y  =  { (/) }  ->  ( A. z ( z  e.  y  ->  -.  z  e.  A )  ->  -.  (/)  e.  A ) )
188prid1 3712 . . . . . . . . . 10  |-  (/)  e.  { (/)
,  { (/) } }
19 eqeq1 2195 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( x  =  { (/) }  <->  (/)  =  { (/)
} ) )
20 eqeq1 2195 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
2120anbi1d 465 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( ( x  =  (/)  /\  ph ) 
<->  ( (/)  =  (/)  /\  ph ) ) )
2219, 21orbi12d 794 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( ( x  =  { (/) }  \/  ( x  =  (/)  /\  ph ) )  <-> 
( (/)  =  { (/) }  \/  ( (/)  =  (/)  /\ 
ph ) ) ) )
2322, 5elrab2 2910 . . . . . . . . . 10  |-  ( (/)  e.  A  <->  ( (/)  e.  { (/)
,  { (/) } }  /\  ( (/)  =  { (/)
}  \/  ( (/)  =  (/)  /\  ph )
) ) )
2418, 23mpbiran 941 . . . . . . . . 9  |-  ( (/)  e.  A  <->  ( (/)  =  { (/)
}  \/  ( (/)  =  (/)  /\  ph )
) )
25 pm2.46 740 . . . . . . . . 9  |-  ( -.  ( (/)  =  { (/)
}  \/  ( (/)  =  (/)  /\  ph )
)  ->  -.  ( (/)  =  (/)  /\  ph )
)
2624, 25sylnbi 679 . . . . . . . 8  |-  ( -.  (/)  e.  A  ->  -.  ( (/)  =  (/)  /\  ph ) )
27 eqid 2188 . . . . . . . . 9  |-  (/)  =  (/)
2827biantrur 303 . . . . . . . 8  |-  ( ph  <->  (
(/)  =  (/)  /\  ph ) )
2926, 28sylnibr 678 . . . . . . 7  |-  ( -.  (/)  e.  A  ->  -.  ph )
3029olcd 735 . . . . . 6  |-  ( -.  (/)  e.  A  ->  ( ph  \/  -.  ph )
)
3117, 30syl6 33 . . . . 5  |-  ( y  =  { (/) }  ->  ( A. z ( z  e.  y  ->  -.  z  e.  A )  ->  ( ph  \/  -.  ph ) ) )
32 orc 713 . . . . . . 7  |-  ( ph  ->  ( ph  \/  -.  ph ) )
3332adantl 277 . . . . . 6  |-  ( ( y  =  (/)  /\  ph )  ->  ( ph  \/  -.  ph ) )
3433a1d 22 . . . . 5  |-  ( ( y  =  (/)  /\  ph )  ->  ( A. z
( z  e.  y  ->  -.  z  e.  A )  ->  ( ph  \/  -.  ph )
) )
3531, 34jaoi 717 . . . 4  |-  ( ( y  =  { (/) }  \/  ( y  =  (/)  /\  ph ) )  ->  ( A. z
( z  e.  y  ->  -.  z  e.  A )  ->  ( ph  \/  -.  ph )
) )
367, 35syl 14 . . 3  |-  ( y  e.  A  ->  ( A. z ( z  e.  y  ->  -.  z  e.  A )  ->  ( ph  \/  -.  ph )
) )
3736imp 124 . 2  |-  ( ( y  e.  A  /\  A. z ( z  e.  y  ->  -.  z  e.  A ) )  -> 
( ph  \/  -.  ph ) )
3837exlimiv 1608 1  |-  ( E. y ( y  e.  A  /\  A. z
( z  e.  y  ->  -.  z  e.  A ) )  -> 
( ph  \/  -.  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 709   A.wal 1361    = wceq 1363   E.wex 1502    e. wcel 2159   {crab 2471   (/)c0 3436   {csn 3606   {cpr 3607
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170  ax-nul 4143
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-rab 2476  df-v 2753  df-dif 3145  df-un 3147  df-nul 3437  df-sn 3612  df-pr 3613
This theorem is referenced by:  regexmid  4548  nnregexmid  4634
  Copyright terms: Public domain W3C validator