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Theorem regexmidlemm 4417
Description: Lemma for regexmid 4420. 
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 4082 . . . 4  |-  { (/) }  e.  _V
21prid2 3600 . . 3  |-  { (/) }  e.  { (/) ,  { (/)
} }
3 eqid 2117 . . . 4  |-  { (/) }  =  { (/) }
43orci 705 . . 3  |-  ( {
(/) }  =  { (/)
}  \/  ( {
(/) }  =  (/)  /\  ph ) )
5 eqeq1 2124 . . . . 5  |-  ( x  =  { (/) }  ->  ( x  =  { (/) }  <->  { (/) }  =  { (/)
} ) )
6 eqeq1 2124 . . . . . 6  |-  ( x  =  { (/) }  ->  ( x  =  (/)  <->  { (/) }  =  (/) ) )
76anbi1d 460 . . . . 5  |-  ( x  =  { (/) }  ->  ( ( x  =  (/)  /\ 
ph )  <->  ( { (/)
}  =  (/)  /\  ph ) ) )
85, 7orbi12d 767 . . . 4  |-  ( x  =  { (/) }  ->  ( ( x  =  { (/)
}  \/  ( x  =  (/)  /\  ph )
)  <->  ( { (/) }  =  { (/) }  \/  ( { (/) }  =  (/)  /\ 
ph ) ) ) )
9 regexmidlemm.a . . . 4  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
108, 9elrab2 2816 . . 3  |-  ( {
(/) }  e.  A  <->  ( { (/) }  e.  { (/)
,  { (/) } }  /\  ( { (/) }  =  { (/) }  \/  ( { (/) }  =  (/)  /\ 
ph ) ) ) )
112, 4, 10mpbir2an 911 . 2  |-  { (/) }  e.  A
12 elex2 2676 . 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 682    = wceq 1316   E.wex 1453    e. wcel 1465   {crab 2397   (/)c0 3333   {csn 3497   {cpr 3498
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504
This theorem is referenced by:  regexmid  4420  reg2exmid  4421  reg3exmid  4464  nnregexmid  4504
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