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Theorem regexmidlemm 4490
Description: Lemma for regexmid 4493. 
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 4149 . . . 4  |-  { (/) }  e.  _V
21prid2 3666 . . 3  |-  { (/) }  e.  { (/) ,  { (/)
} }
3 eqid 2157 . . . 4  |-  { (/) }  =  { (/) }
43orci 721 . . 3  |-  ( {
(/) }  =  { (/)
}  \/  ( {
(/) }  =  (/)  /\  ph ) )
5 eqeq1 2164 . . . . 5  |-  ( x  =  { (/) }  ->  ( x  =  { (/) }  <->  { (/) }  =  { (/)
} ) )
6 eqeq1 2164 . . . . . 6  |-  ( x  =  { (/) }  ->  ( x  =  (/)  <->  { (/) }  =  (/) ) )
76anbi1d 461 . . . . 5  |-  ( x  =  { (/) }  ->  ( ( x  =  (/)  /\ 
ph )  <->  ( { (/)
}  =  (/)  /\  ph ) ) )
85, 7orbi12d 783 . . . 4  |-  ( x  =  { (/) }  ->  ( ( x  =  { (/)
}  \/  ( x  =  (/)  /\  ph )
)  <->  ( { (/) }  =  { (/) }  \/  ( { (/) }  =  (/)  /\ 
ph ) ) ) )
9 regexmidlemm.a . . . 4  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
108, 9elrab2 2871 . . 3  |-  ( {
(/) }  e.  A  <->  ( { (/) }  e.  { (/)
,  { (/) } }  /\  ( { (/) }  =  { (/) }  \/  ( { (/) }  =  (/)  /\ 
ph ) ) ) )
112, 4, 10mpbir2an 927 . 2  |-  { (/) }  e.  A
12 elex2 2728 . 2  |-  ( {
(/) }  e.  A  ->  E. y  y  e.  A )
1311, 12ax-mp 5 1  |-  E. y 
y  e.  A
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 698    = wceq 1335   E.wex 1472    e. wcel 2128   {crab 2439   (/)c0 3394   {csn 3560   {cpr 3561
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567
This theorem is referenced by:  regexmid  4493  reg2exmid  4494  reg3exmid  4538  nnregexmid  4579
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