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Theorem regexmidlem1 4286
Description: Lemma for regexmid 4288. 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 2062 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  { (/) }  <-> 
y  =  { (/) } ) )
2 eqeq1 2062 . . . . . . . 8  |-  ( x  =  y  ->  (
x  =  (/)  <->  y  =  (/) ) )
32anbi1d 446 . . . . . . 7  |-  ( x  =  y  ->  (
( x  =  (/)  /\ 
ph )  <->  ( y  =  (/)  /\  ph )
) )
41, 3orbi12d 717 . . . . . 6  |-  ( x  =  y  ->  (
( x  =  { (/)
}  \/  ( x  =  (/)  /\  ph )
)  <->  ( y  =  { (/) }  \/  (
y  =  (/)  /\  ph ) ) ) )
5 regexmidlemm.a . . . . . 6  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
64, 5elrab2 2723 . . . . 5  |-  ( y  e.  A  <->  ( y  e.  { (/) ,  { (/) } }  /\  ( y  =  { (/) }  \/  ( y  =  (/)  /\ 
ph ) ) ) )
76simprbi 264 . . . 4  |-  ( y  e.  A  ->  (
y  =  { (/) }  \/  ( y  =  (/)  /\  ph ) ) )
8 0ex 3912 . . . . . . . . 9  |-  (/)  e.  _V
98snid 3430 . . . . . . . 8  |-  (/)  e.  { (/)
}
10 eleq2 2117 . . . . . . . 8  |-  ( y  =  { (/) }  ->  (
(/)  e.  y  <->  (/)  e.  { (/)
} ) )
119, 10mpbiri 161 . . . . . . 7  |-  ( y  =  { (/) }  ->  (/)  e.  y )
12 eleq1 2116 . . . . . . . . 9  |-  ( z  =  (/)  ->  ( z  e.  y  <->  (/)  e.  y ) )
13 eleq1 2116 . . . . . . . . . 10  |-  ( z  =  (/)  ->  ( z  e.  A  <->  (/)  e.  A
) )
1413notbid 602 . . . . . . . . 9  |-  ( z  =  (/)  ->  ( -.  z  e.  A  <->  -.  (/)  e.  A
) )
1512, 14imbi12d 227 . . . . . . . 8  |-  ( z  =  (/)  ->  ( ( z  e.  y  ->  -.  z  e.  A
)  <->  ( (/)  e.  y  ->  -.  (/)  e.  A
) ) )
168, 15spcv 2663 . . . . . . 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 3504 . . . . . . . . . 10  |-  (/)  e.  { (/)
,  { (/) } }
19 eqeq1 2062 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( x  =  { (/) }  <->  (/)  =  { (/)
} ) )
20 eqeq1 2062 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
2120anbi1d 446 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( ( x  =  (/)  /\  ph ) 
<->  ( (/)  =  (/)  /\  ph ) ) )
2219, 21orbi12d 717 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( ( x  =  { (/) }  \/  ( x  =  (/)  /\  ph ) )  <-> 
( (/)  =  { (/) }  \/  ( (/)  =  (/)  /\ 
ph ) ) ) )
2322, 5elrab2 2723 . . . . . . . . . 10  |-  ( (/)  e.  A  <->  ( (/)  e.  { (/)
,  { (/) } }  /\  ( (/)  =  { (/)
}  \/  ( (/)  =  (/)  /\  ph )
) ) )
2418, 23mpbiran 858 . . . . . . . . 9  |-  ( (/)  e.  A  <->  ( (/)  =  { (/)
}  \/  ( (/)  =  (/)  /\  ph )
) )
25 pm2.46 668 . . . . . . . . 9  |-  ( -.  ( (/)  =  { (/)
}  \/  ( (/)  =  (/)  /\  ph )
)  ->  -.  ( (/)  =  (/)  /\  ph )
)
2624, 25sylnbi 613 . . . . . . . 8  |-  ( -.  (/)  e.  A  ->  -.  ( (/)  =  (/)  /\  ph ) )
27 eqid 2056 . . . . . . . . 9  |-  (/)  =  (/)
2827biantrur 291 . . . . . . . 8  |-  ( ph  <->  (
(/)  =  (/)  /\  ph ) )
2926, 28sylnibr 612 . . . . . . 7  |-  ( -.  (/)  e.  A  ->  -.  ph )
3029olcd 663 . . . . . 6  |-  ( -.  (/)  e.  A  ->  ( ph  \/  -.  ph )
)
3117, 30syl6 33 . . . . 5  |-  ( y  =  { (/) }  ->  ( A. z ( z  e.  y  ->  -.  z  e.  A )  ->  ( ph  \/  -.  ph ) ) )
32 orc 643 . . . . . . 7  |-  ( ph  ->  ( ph  \/  -.  ph ) )
3332adantl 266 . . . . . 6  |-  ( ( y  =  (/)  /\  ph )  ->  ( ph  \/  -.  ph ) )
3433a1d 22 . . . . 5  |-  ( ( y  =  (/)  /\  ph )  ->  ( A. z
( z  e.  y  ->  -.  z  e.  A )  ->  ( ph  \/  -.  ph )
) )
3531, 34jaoi 646 . . . 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 119 . 2  |-  ( ( y  e.  A  /\  A. z ( z  e.  y  ->  -.  z  e.  A ) )  -> 
( ph  \/  -.  ph ) )
3837exlimiv 1505 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 101    \/ wo 639   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   {crab 2327   (/)c0 3252   {csn 3403   {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3911
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-nul 3253  df-sn 3409  df-pr 3410
This theorem is referenced by:  regexmid  4288  nnregexmid  4370
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