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Theorem reg2exmidlema 4444
Description: Lemma for reg2exmid 4446. 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 519 . . . . . . 7  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  u  =  { (/) } )  ->  A. v  e.  A  u  C_  v )
2 sseq1 3115 . . . . . . . . 9  |-  ( u  =  { (/) }  ->  ( u  C_  v  <->  { (/) }  C_  v ) )
32ralbidv 2435 . . . . . . . 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 4050 . . . . . . . 8  |-  (/)  e.  _V
76snss 3644 . . . . . . 7  |-  ( (/)  e.  v  <->  { (/) }  C_  v
)
87ralbii 2439 . . . . . 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 3362 . . . . . 6  |-  -.  (/)  e.  (/)
11 eqid 2137 . . . . . . . . . . . 12  |-  (/)  =  (/)
1211jctl 312 . . . . . . . . . . 11  |-  ( ph  ->  ( (/)  =  (/)  /\  ph ) )
1312olcd 723 . . . . . . . . . 10  |-  ( ph  ->  ( (/)  =  { (/)
}  \/  ( (/)  =  (/)  /\  ph )
) )
146prid1 3624 . . . . . . . . . 10  |-  (/)  e.  { (/)
,  { (/) } }
1513, 14jctil 310 . . . . . . . . 9  |-  ( ph  ->  ( (/)  e.  { (/) ,  { (/) } }  /\  ( (/)  =  { (/) }  \/  ( (/)  =  (/)  /\ 
ph ) ) ) )
16 eqeq1 2144 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( x  =  { (/) }  <->  (/)  =  { (/)
} ) )
17 eqeq1 2144 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
1817anbi1d 460 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( ( x  =  (/)  /\  ph ) 
<->  ( (/)  =  (/)  /\  ph ) ) )
1916, 18orbi12d 782 . . . . . . . . . 10  |-  ( x  =  (/)  ->  ( ( x  =  { (/) }  \/  ( x  =  (/)  /\  ph ) )  <-> 
( (/)  =  { (/) }  \/  ( (/)  =  (/)  /\ 
ph ) ) ) )
20 regexmidlemm.a . . . . . . . . . 10  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
2119, 20elrab2 2838 . . . . . . . . 9  |-  ( (/)  e.  A  <->  ( (/)  e.  { (/)
,  { (/) } }  /\  ( (/)  =  { (/)
}  \/  ( (/)  =  (/)  /\  ph )
) ) )
2215, 21sylibr 133 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  A )
23 eleq2 2201 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( (/)  e.  v  <->  (/)  e.  (/) ) )
2423rspcv 2780 . . . . . . . 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 653 . . . . 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 723 . . 3  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  u  =  { (/) } )  ->  ( ph  \/  -.  ph ) )
30 simprr 521 . . . 4  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  ( u  =  (/)  /\  ph ) )  ->  ph )
3130orcd 722 . . 3  |-  ( ( ( u  e.  A  /\  A. v  e.  A  u  C_  v )  /\  ( u  =  (/)  /\  ph ) )  ->  ( ph  \/  -.  ph )
)
32 eqeq1 2144 . . . . . . 7  |-  ( x  =  u  ->  (
x  =  { (/) }  <-> 
u  =  { (/) } ) )
33 eqeq1 2144 . . . . . . . 8  |-  ( x  =  u  ->  (
x  =  (/)  <->  u  =  (/) ) )
3433anbi1d 460 . . . . . . 7  |-  ( x  =  u  ->  (
( x  =  (/)  /\ 
ph )  <->  ( u  =  (/)  /\  ph )
) )
3532, 34orbi12d 782 . . . . . 6  |-  ( x  =  u  ->  (
( x  =  { (/)
}  \/  ( x  =  (/)  /\  ph )
)  <->  ( u  =  { (/) }  \/  (
u  =  (/)  /\  ph ) ) ) )
3635, 20elrab2 2838 . . . . 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 787 . 2  |-  ( ( u  e.  A  /\  A. v  e.  A  u 
C_  v )  -> 
( ph  \/  -.  ph ) )
4039rexlimiva 2542 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 697    = wceq 1331    e. wcel 1480   A.wral 2414   E.wrex 2415   {crab 2418    C_ wss 3066   (/)c0 3358   {csn 3522   {cpr 3523
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-nul 4049
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-sn 3528  df-pr 3529
This theorem is referenced by:  reg2exmid  4446  reg3exmid  4489
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