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Theorem onsucelsucexmidlem1 4307
Description: Lemma for onsucelsucexmid 4309. (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 3931 . . 3  |-  (/)  e.  _V
21prid1 3522 . 2  |-  (/)  e.  { (/)
,  { (/) } }
3 eqid 2083 . . 3  |-  (/)  =  (/)
43orci 683 . 2  |-  ( (/)  =  (/)  \/  ph )
5 eqeq1 2089 . . . 4  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
65orbi1d 738 . . 3  |-  ( x  =  (/)  ->  ( ( x  =  (/)  \/  ph ) 
<->  ( (/)  =  (/)  \/  ph ) ) )
76elrab 2759 . 2  |-  ( (/)  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  <-> 
( (/)  e.  { (/) ,  { (/) } }  /\  ( (/)  =  (/)  \/  ph ) ) )
82, 4, 7mpbir2an 884 1  |-  (/)  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 662    = wceq 1285    e. wcel 1434   {crab 2357   (/)c0 3269   {csn 3422   {cpr 3423
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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-nul 3930
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-nul 3270  df-sn 3428  df-pr 3429
This theorem is referenced by:  onsucelsucexmidlem  4308  onsucelsucexmid  4309  acexmidlem2  5588
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