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Theorem reg2exmidlema 4350
Description: Lemma for reg2exmid 4352. 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 497 . . . . . . 7  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  u  =  { (/) } )  ->  A. v  e.  A  u  C_  v )
2 sseq1 3047 . . . . . . . . 9  |-  ( u  =  { (/) }  ->  ( u  C_  v  <->  { (/) }  C_  v ) )
32ralbidv 2380 . . . . . . . 8  |-  ( u  =  { (/) }  ->  ( A. v  e.  A  u  C_  v  <->  A. v  e.  A  { (/) }  C_  v ) )
43adantl 271 . . . . . . 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 145 . . . . . 6  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  u  =  { (/) } )  ->  A. v  e.  A  { (/) }  C_  v
)
6 0ex 3966 . . . . . . . 8  |-  (/)  e.  _V
76snss 3566 . . . . . . 7  |-  ( (/)  e.  v  <->  { (/) }  C_  v
)
87ralbii 2384 . . . . . 6  |-  ( A. v  e.  A  (/)  e.  v  <->  A. v  e.  A  { (/) }  C_  v
)
95, 8sylibr 132 . . . . 5  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  u  =  { (/) } )  ->  A. v  e.  A  (/) 
e.  v )
10 noel 3290 . . . . . 6  |-  -.  (/)  e.  (/)
11 eqid 2088 . . . . . . . . . . . 12  |-  (/)  =  (/)
1211jctl 307 . . . . . . . . . . 11  |-  ( ph  ->  ( (/)  =  (/)  /\  ph ) )
1312olcd 688 . . . . . . . . . 10  |-  ( ph  ->  ( (/)  =  { (/)
}  \/  ( (/)  =  (/)  /\  ph )
) )
146prid1 3548 . . . . . . . . . 10  |-  (/)  e.  { (/)
,  { (/) } }
1513, 14jctil 305 . . . . . . . . 9  |-  ( ph  ->  ( (/)  e.  { (/) ,  { (/) } }  /\  ( (/)  =  { (/) }  \/  ( (/)  =  (/)  /\ 
ph ) ) ) )
16 eqeq1 2094 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( x  =  { (/) }  <->  (/)  =  { (/)
} ) )
17 eqeq1 2094 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
1817anbi1d 453 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( ( x  =  (/)  /\  ph ) 
<->  ( (/)  =  (/)  /\  ph ) ) )
1916, 18orbi12d 742 . . . . . . . . . 10  |-  ( x  =  (/)  ->  ( ( x  =  { (/) }  \/  ( x  =  (/)  /\  ph ) )  <-> 
( (/)  =  { (/) }  \/  ( (/)  =  (/)  /\ 
ph ) ) ) )
20 regexmidlemm.a . . . . . . . . . 10  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
2119, 20elrab2 2774 . . . . . . . . 9  |-  ( (/)  e.  A  <->  ( (/)  e.  { (/)
,  { (/) } }  /\  ( (/)  =  { (/)
}  \/  ( (/)  =  (/)  /\  ph )
) ) )
2215, 21sylibr 132 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  A )
23 eleq2 2151 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( (/)  e.  v  <->  (/)  e.  (/) ) )
2423rspcv 2718 . . . . . . . 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 625 . . . . 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 688 . . 3  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  u  =  { (/) } )  ->  ( ph  \/  -.  ph ) )
30 simprr 499 . . . 4  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  ( u  =  (/)  /\  ph ) )  ->  ph )
3130orcd 687 . . 3  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  ( u  =  (/)  /\  ph ) )  ->  ( ph  \/  -.  ph )
)
32 eqeq1 2094 . . . . . . 7  |-  ( x  =  u  ->  (
x  =  { (/) }  <-> 
u  =  { (/) } ) )
33 eqeq1 2094 . . . . . . . 8  |-  ( x  =  u  ->  (
x  =  (/)  <->  u  =  (/) ) )
3433anbi1d 453 . . . . . . 7  |-  ( x  =  u  ->  (
( x  =  (/)  /\ 
ph )  <->  ( u  =  (/)  /\  ph )
) )
3532, 34orbi12d 742 . . . . . 6  |-  ( x  =  u  ->  (
( x  =  { (/)
}  \/  ( x  =  (/)  /\  ph )
)  <->  ( u  =  { (/) }  \/  (
u  =  (/)  /\  ph ) ) ) )
3635, 20elrab2 2774 . . . . 5  |-  ( u  e.  A  <->  ( u  e.  { (/) ,  { (/) } }  /\  ( u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) ) )
3736simprbi 269 . . . 4  |-  ( u  e.  A  ->  (
u  =  { (/) }  \/  ( u  =  (/)  /\  ph ) ) )
3837adantr 270 . . 3  |-  ( ( u  e.  A  /\  A. v  e.  A  u 
C_  v )  -> 
( u  =  { (/)
}  \/  ( u  =  (/)  /\  ph )
) )
3929, 31, 38mpjaodan 747 . 2  |-  ( ( u  e.  A  /\  A. v  e.  A  u 
C_  v )  -> 
( ph  \/  -.  ph ) )
4039rexlimiva 2484 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 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363    C_ wss 2999   (/)c0 3286   {csn 3446   {cpr 3447
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3965
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3452  df-pr 3453
This theorem is referenced by:  reg2exmid  4352  reg3exmid  4395
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