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Theorem regexmidlemm 4283
Description: Lemma for regexmid 4286. 
A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
Hypothesis
Ref Expression
regexmidlemm.a  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
Assertion
Ref Expression
regexmidlemm  |-  E. y 
y  e.  A
Distinct variable groups:    y, A    ph, x, y
Allowed substitution hint:    A( x)

Proof of Theorem regexmidlemm
StepHypRef Expression
1 p0ex 3967 . . . 4  |-  { (/) }  e.  _V
21prid2 3507 . . 3  |-  { (/) }  e.  { (/) ,  { (/)
} }
3 eqid 2082 . . . 4  |-  { (/) }  =  { (/) }
43orci 683 . . 3  |-  ( {
(/) }  =  { (/)
}  \/  ( {
(/) }  =  (/)  /\  ph ) )
5 eqeq1 2088 . . . . 5  |-  ( x  =  { (/) }  ->  ( x  =  { (/) }  <->  { (/) }  =  { (/)
} ) )
6 eqeq1 2088 . . . . . 6  |-  ( x  =  { (/) }  ->  ( x  =  (/)  <->  { (/) }  =  (/) ) )
76anbi1d 453 . . . . 5  |-  ( x  =  { (/) }  ->  ( ( x  =  (/)  /\ 
ph )  <->  ( { (/)
}  =  (/)  /\  ph ) ) )
85, 7orbi12d 740 . . . 4  |-  ( x  =  { (/) }  ->  ( ( x  =  { (/)
}  \/  ( x  =  (/)  /\  ph )
)  <->  ( { (/) }  =  { (/) }  \/  ( { (/) }  =  (/)  /\ 
ph ) ) ) )
9 regexmidlemm.a . . . 4  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
108, 9elrab2 2752 . . 3  |-  ( {
(/) }  e.  A  <->  ( { (/) }  e.  { (/)
,  { (/) } }  /\  ( { (/) }  =  { (/) }  \/  ( { (/) }  =  (/)  /\ 
ph ) ) ) )
112, 4, 10mpbir2an 884 . 2  |-  { (/) }  e.  A
12 elex2 2616 . 2  |-  ( {
(/) }  e.  A  ->  E. y  y  e.  A )
1311, 12ax-mp 7 1  |-  E. y 
y  e.  A
Colors of variables: wff set class
Syntax hints:    /\ wa 102    \/ wo 662    = wceq 1285   E.wex 1422    e. wcel 1434   {crab 2353   (/)c0 3258   {csn 3406   {cpr 3407
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-nul 3912  ax-pow 3956
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  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
This theorem is referenced by:  regexmid  4286  reg2exmid  4287  reg3exmid  4330  nnregexmid  4368
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