ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  onsucelsucexmidlem1 Unicode version

Theorem onsucelsucexmidlem1 4545
Description: Lemma for onsucelsucexmid 4547. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
onsucelsucexmidlem1  |-  (/)  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
Distinct variable group:    ph, x

Proof of Theorem onsucelsucexmidlem1
StepHypRef Expression
1 0ex 4145 . . 3  |-  (/)  e.  _V
21prid1 3713 . 2  |-  (/)  e.  { (/)
,  { (/) } }
3 eqid 2189 . . 3  |-  (/)  =  (/)
43orci 732 . 2  |-  ( (/)  =  (/)  \/  ph )
5 eqeq1 2196 . . . 4  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
65orbi1d 792 . . 3  |-  ( x  =  (/)  ->  ( ( x  =  (/)  \/  ph ) 
<->  ( (/)  =  (/)  \/  ph ) ) )
76elrab 2908 . 2  |-  ( (/)  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  <-> 
( (/)  e.  { (/) ,  { (/) } }  /\  ( (/)  =  (/)  \/  ph ) ) )
82, 4, 7mpbir2an 944 1  |-  (/)  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 709    = wceq 1364    e. wcel 2160   {crab 2472   (/)c0 3437   {csn 3607   {cpr 3608
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-ext 2171  ax-nul 4144
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 2754  df-dif 3146  df-un 3148  df-nul 3438  df-sn 3613  df-pr 3614
This theorem is referenced by:  onsucelsucexmidlem  4546  onsucelsucexmid  4547  acexmidlem2  5892
  Copyright terms: Public domain W3C validator