ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  regexmidlemm Unicode version

Theorem regexmidlemm 4556
Description: Lemma for regexmid 4559. 
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 4213 . . . 4  |-  { (/) }  e.  _V
21prid2 3721 . . 3  |-  { (/) }  e.  { (/) ,  { (/)
} }
3 eqid 2189 . . . 4  |-  { (/) }  =  { (/) }
43orci 732 . . 3  |-  ( {
(/) }  =  { (/)
}  \/  ( {
(/) }  =  (/)  /\  ph ) )
5 eqeq1 2196 . . . . 5  |-  ( x  =  { (/) }  ->  ( x  =  { (/) }  <->  { (/) }  =  { (/)
} ) )
6 eqeq1 2196 . . . . . 6  |-  ( x  =  { (/) }  ->  ( x  =  (/)  <->  { (/) }  =  (/) ) )
76anbi1d 465 . . . . 5  |-  ( x  =  { (/) }  ->  ( ( x  =  (/)  /\ 
ph )  <->  ( { (/)
}  =  (/)  /\  ph ) ) )
85, 7orbi12d 794 . . . 4  |-  ( x  =  { (/) }  ->  ( ( x  =  { (/)
}  \/  ( x  =  (/)  /\  ph )
)  <->  ( { (/) }  =  { (/) }  \/  ( { (/) }  =  (/)  /\ 
ph ) ) ) )
9 regexmidlemm.a . . . 4  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  (
x  =  (/)  /\  ph ) ) }
108, 9elrab2 2915 . . 3  |-  ( {
(/) }  e.  A  <->  ( { (/) }  e.  { (/)
,  { (/) } }  /\  ( { (/) }  =  { (/) }  \/  ( { (/) }  =  (/)  /\ 
ph ) ) ) )
112, 4, 10mpbir2an 944 . 2  |-  { (/) }  e.  A
12 elex2 2772 . 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 104    \/ wo 709    = wceq 1364   E.wex 1503    e. wcel 2160   {crab 2472   (/)c0 3442   {csn 3614   {cpr 3615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-nul 4151  ax-pow 4199
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2758  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-nul 3443  df-pw 3599  df-sn 3620  df-pr 3621
This theorem is referenced by:  regexmid  4559  reg2exmid  4560  reg3exmid  4604  nnregexmid  4645
  Copyright terms: Public domain W3C validator