ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reg3exmidlemwe Unicode version

Theorem reg3exmidlemwe 4329
Description: Lemma for reg3exmid 4330. Our counterexample  A satisfies  We. (Contributed by Jim Kingdon, 3-Oct-2021.)
Hypothesis
Ref Expression
reg3exmidlemwe.a  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
Assertion
Ref Expression
reg3exmidlemwe  |-  _E  We  A
Distinct variable group:    ph, x
Allowed substitution hint:    A( x)

Proof of Theorem reg3exmidlemwe
Dummy variables  a  b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zfregfr 4324 . 2  |-  _E  Fr  A
2 epel 4055 . . . . . 6  |-  ( a  _E  b  <->  a  e.  b )
3 epel 4055 . . . . . 6  |-  ( b  _E  c  <->  b  e.  c )
42, 3anbi12i 448 . . . . 5  |-  ( ( a  _E  b  /\  b  _E  c )  <->  ( a  e.  b  /\  b  e.  c )
)
5 simpr 108 . . . . . 6  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  e.  b  /\  b  e.  c ) )  -> 
( a  e.  b  /\  b  e.  c ) )
6 elirr 4292 . . . . . . . 8  |-  -.  { (/)
}  e.  { (/) }
7 simprr 499 . . . . . . . . . 10  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  e.  b  /\  b  e.  c ) )  -> 
b  e.  c )
8 noel 3262 . . . . . . . . . . . . 13  |-  -.  a  e.  (/)
9 eleq2 2143 . . . . . . . . . . . . 13  |-  ( b  =  (/)  ->  ( a  e.  b  <->  a  e.  (/) ) )
108, 9mtbiri 633 . . . . . . . . . . . 12  |-  ( b  =  (/)  ->  -.  a  e.  b )
11 simprl 498 . . . . . . . . . . . 12  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  e.  b  /\  b  e.  c ) )  -> 
a  e.  b )
1210, 11nsyl3 589 . . . . . . . . . . 11  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  e.  b  /\  b  e.  c ) )  ->  -.  b  =  (/) )
13 elrabi 2747 . . . . . . . . . . . . . . . 16  |-  ( b  e.  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  { (/) }  \/  ( x  =  (/)  /\  ph ) ) }  ->  b  e.  { (/) ,  { (/)
} } )
14 reg3exmidlemwe.a . . . . . . . . . . . . . . . 16  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
1513, 14eleq2s 2174 . . . . . . . . . . . . . . 15  |-  ( b  e.  A  ->  b  e.  { (/) ,  { (/) } } )
16 elpri 3429 . . . . . . . . . . . . . . 15  |-  ( b  e.  { (/) ,  { (/)
} }  ->  (
b  =  (/)  \/  b  =  { (/) } ) )
1715, 16syl 14 . . . . . . . . . . . . . 14  |-  ( b  e.  A  ->  (
b  =  (/)  \/  b  =  { (/) } ) )
1817orcomd 681 . . . . . . . . . . . . 13  |-  ( b  e.  A  ->  (
b  =  { (/) }  \/  b  =  (/) ) )
19183ad2ant2 961 . . . . . . . . . . . 12  |-  ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A )  ->  ( b  =  { (/)
}  \/  b  =  (/) ) )
2019adantr 270 . . . . . . . . . . 11  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  e.  b  /\  b  e.  c ) )  -> 
( b  =  { (/)
}  \/  b  =  (/) ) )
2112, 20ecased 1281 . . . . . . . . . 10  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  e.  b  /\  b  e.  c ) )  -> 
b  =  { (/) } )
22 noel 3262 . . . . . . . . . . . . 13  |-  -.  b  e.  (/)
23 eleq2 2143 . . . . . . . . . . . . 13  |-  ( c  =  (/)  ->  ( b  e.  c  <->  b  e.  (/) ) )
2422, 23mtbiri 633 . . . . . . . . . . . 12  |-  ( c  =  (/)  ->  -.  b  e.  c )
2524, 7nsyl3 589 . . . . . . . . . . 11  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  e.  b  /\  b  e.  c ) )  ->  -.  c  =  (/) )
26 elrabi 2747 . . . . . . . . . . . . . . . 16  |-  ( c  e.  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  { (/) }  \/  ( x  =  (/)  /\  ph ) ) }  ->  c  e.  { (/) ,  { (/)
} } )
2726, 14eleq2s 2174 . . . . . . . . . . . . . . 15  |-  ( c  e.  A  ->  c  e.  { (/) ,  { (/) } } )
28 vex 2605 . . . . . . . . . . . . . . . 16  |-  c  e. 
_V
2928elpr 3427 . . . . . . . . . . . . . . 15  |-  ( c  e.  { (/) ,  { (/)
} }  <->  ( c  =  (/)  \/  c  =  { (/) } ) )
3027, 29sylib 120 . . . . . . . . . . . . . 14  |-  ( c  e.  A  ->  (
c  =  (/)  \/  c  =  { (/) } ) )
3130orcomd 681 . . . . . . . . . . . . 13  |-  ( c  e.  A  ->  (
c  =  { (/) }  \/  c  =  (/) ) )
32313ad2ant3 962 . . . . . . . . . . . 12  |-  ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A )  ->  ( c  =  { (/)
}  \/  c  =  (/) ) )
3332adantr 270 . . . . . . . . . . 11  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  e.  b  /\  b  e.  c ) )  -> 
( c  =  { (/)
}  \/  c  =  (/) ) )
3425, 33ecased 1281 . . . . . . . . . 10  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  e.  b  /\  b  e.  c ) )  -> 
c  =  { (/) } )
357, 21, 343eltr3d 2162 . . . . . . . . 9  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  e.  b  /\  b  e.  c ) )  ->  { (/) }  e.  { (/)
} )
3635ex 113 . . . . . . . 8  |-  ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A )  ->  ( ( a  e.  b  /\  b  e.  c )  ->  { (/) }  e.  { (/) } ) )
376, 36mtoi 623 . . . . . . 7  |-  ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A )  ->  -.  ( a  e.  b  /\  b  e.  c ) )
3837adantr 270 . . . . . 6  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  e.  b  /\  b  e.  c ) )  ->  -.  ( a  e.  b  /\  b  e.  c ) )
395, 38pm2.21dd 583 . . . . 5  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  e.  b  /\  b  e.  c ) )  -> 
a  _E  c )
404, 39sylan2b 281 . . . 4  |-  ( ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A
)  /\  ( a  _E  b  /\  b  _E  c ) )  -> 
a  _E  c )
4140ex 113 . . 3  |-  ( ( a  e.  A  /\  b  e.  A  /\  c  e.  A )  ->  ( ( a  _E  b  /\  b  _E  c )  ->  a  _E  c ) )
4241rgen3 2449 . 2  |-  A. a  e.  A  A. b  e.  A  A. c  e.  A  ( (
a  _E  b  /\  b  _E  c )  ->  a  _E  c )
43 df-wetr 4097 . 2  |-  (  _E  We  A  <->  (  _E  Fr  A  /\  A. a  e.  A  A. b  e.  A  A. c  e.  A  ( (
a  _E  b  /\  b  _E  c )  ->  a  _E  c ) ) )
441, 42, 43mpbir2an 884 1  |-  _E  We  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2349   {crab 2353   (/)c0 3258   {csn 3406   {cpr 3407   class class class wbr 3793    _E cep 4050    Fr wfr 4091    We wwe 4093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-eprel 4052  df-frfor 4094  df-frind 4095  df-wetr 4097
This theorem is referenced by:  reg3exmid  4330
  Copyright terms: Public domain W3C validator