ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reg2exmidlema Unicode version

Theorem reg2exmidlema 4419
Description: Lemma for reg2exmid 4421. If  A has a minimal element (expressed by  C_), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
Hypothesis
Ref Expression
regexmidlemm.a  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
Assertion
Ref Expression
reg2exmidlema  |-  ( E. u  e.  A  A. v  e.  A  u  C_  v  ->  ( ph  \/  -.  ph ) )
Distinct variable groups:    ph, x    v, A    ph, u, x    v, u
Allowed substitution hints:    ph( v)    A( x, u)

Proof of Theorem reg2exmidlema
StepHypRef Expression
1 simplr 504 . . . . . . 7  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  u  =  { (/) } )  ->  A. v  e.  A  u  C_  v )
2 sseq1 3090 . . . . . . . . 9  |-  ( u  =  { (/) }  ->  ( u  C_  v  <->  { (/) }  C_  v ) )
32ralbidv 2414 . . . . . . . 8  |-  ( u  =  { (/) }  ->  ( A. v  e.  A  u  C_  v  <->  A. v  e.  A  { (/) }  C_  v ) )
43adantl 275 . . . . . . 7  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  u  =  { (/) } )  ->  ( A. v  e.  A  u  C_  v  <->  A. v  e.  A  { (/)
}  C_  v )
)
51, 4mpbid 146 . . . . . 6  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  u  =  { (/) } )  ->  A. v  e.  A  { (/) }  C_  v
)
6 0ex 4025 . . . . . . . 8  |-  (/)  e.  _V
76snss 3619 . . . . . . 7  |-  ( (/)  e.  v  <->  { (/) }  C_  v
)
87ralbii 2418 . . . . . 6  |-  ( A. v  e.  A  (/)  e.  v  <->  A. v  e.  A  { (/) }  C_  v
)
95, 8sylibr 133 . . . . 5  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  u  =  { (/) } )  ->  A. v  e.  A  (/) 
e.  v )
10 noel 3337 . . . . . 6  |-  -.  (/)  e.  (/)
11 eqid 2117 . . . . . . . . . . . 12  |-  (/)  =  (/)
1211jctl 312 . . . . . . . . . . 11  |-  ( ph  ->  ( (/)  =  (/)  /\  ph ) )
1312olcd 708 . . . . . . . . . 10  |-  ( ph  ->  ( (/)  =  { (/)
}  \/  ( (/)  =  (/)  /\  ph )
) )
146prid1 3599 . . . . . . . . . 10  |-  (/)  e.  { (/)
,  { (/) } }
1513, 14jctil 310 . . . . . . . . 9  |-  ( ph  ->  ( (/)  e.  { (/) ,  { (/) } }  /\  ( (/)  =  { (/) }  \/  ( (/)  =  (/)  /\ 
ph ) ) ) )
16 eqeq1 2124 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( x  =  { (/) }  <->  (/)  =  { (/)
} ) )
17 eqeq1 2124 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
1817anbi1d 460 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( ( x  =  (/)  /\  ph ) 
<->  ( (/)  =  (/)  /\  ph ) ) )
1916, 18orbi12d 767 . . . . . . . . . 10  |-  ( x  =  (/)  ->  ( ( x  =  { (/) }  \/  ( x  =  (/)  /\  ph ) )  <-> 
( (/)  =  { (/) }  \/  ( (/)  =  (/)  /\ 
ph ) ) ) )
20 regexmidlemm.a . . . . . . . . . 10  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
2119, 20elrab2 2816 . . . . . . . . 9  |-  ( (/)  e.  A  <->  ( (/)  e.  { (/)
,  { (/) } }  /\  ( (/)  =  { (/)
}  \/  ( (/)  =  (/)  /\  ph )
) ) )
2215, 21sylibr 133 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  A )
23 eleq2 2181 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( (/)  e.  v  <->  (/)  e.  (/) ) )
2423rspcv 2759 . . . . . . . 8  |-  ( (/)  e.  A  ->  ( A. v  e.  A  (/)  e.  v  ->  (/)  e.  (/) ) )
2522, 24syl 14 . . . . . . 7  |-  ( ph  ->  ( A. v  e.  A  (/)  e.  v  -> 
(/)  e.  (/) ) )
2625com12 30 . . . . . 6  |-  ( A. v  e.  A  (/)  e.  v  ->  ( ph  ->  (/)  e.  (/) ) )
2710, 26mtoi 638 . . . . 5  |-  ( A. v  e.  A  (/)  e.  v  ->  -.  ph )
289, 27syl 14 . . . 4  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  u  =  { (/) } )  ->  -.  ph )
2928olcd 708 . . 3  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  u  =  { (/) } )  ->  ( ph  \/  -.  ph ) )
30 simprr 506 . . . 4  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  ( u  =  (/)  /\  ph ) )  ->  ph )
3130orcd 707 . . 3  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  ( u  =  (/)  /\  ph ) )  ->  ( ph  \/  -.  ph )
)
32 eqeq1 2124 . . . . . . 7  |-  ( x  =  u  ->  (
x  =  { (/) }  <-> 
u  =  { (/) } ) )
33 eqeq1 2124 . . . . . . . 8  |-  ( x  =  u  ->  (
x  =  (/)  <->  u  =  (/) ) )
3433anbi1d 460 . . . . . . 7  |-  ( x  =  u  ->  (
( x  =  (/)  /\ 
ph )  <->  ( u  =  (/)  /\  ph )
) )
3532, 34orbi12d 767 . . . . . 6  |-  ( x  =  u  ->  (
( x  =  { (/)
}  \/  ( x  =  (/)  /\  ph )
)  <->  ( u  =  { (/) }  \/  (
u  =  (/)  /\  ph ) ) ) )
3635, 20elrab2 2816 . . . . 5  |-  ( u  e.  A  <->  ( u  e.  { (/) ,  { (/) } }  /\  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) ) )
3736simprbi 273 . . . 4  |-  ( u  e.  A  ->  (
u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) )
3837adantr 274 . . 3  |-  ( ( u  e.  A  /\  A. v  e.  A  u 
C_  v )  -> 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) )
3929, 31, 38mpjaodan 772 . 2  |-  ( ( u  e.  A  /\  A. v  e.  A  u 
C_  v )  -> 
( ph  \/  -.  ph ) )
4039rexlimiva 2521 1  |-  ( E. u  e.  A  A. v  e.  A  u  C_  v  ->  ( ph  \/  -.  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 682    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   {crab 2397    C_ wss 3041   (/)c0 3333   {csn 3497   {cpr 3498
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-pr 3504
This theorem is referenced by:  reg2exmid  4421  reg3exmid  4464
  Copyright terms: Public domain W3C validator