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Theorem onsucelsucexmidlem1 4381
Description: Lemma for onsucelsucexmid 4383. (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 3995 . . 3  |-  (/)  e.  _V
21prid1 3576 . 2  |-  (/)  e.  { (/)
,  { (/) } }
3 eqid 2100 . . 3  |-  (/)  =  (/)
43orci 691 . 2  |-  ( (/)  =  (/)  \/  ph )
5 eqeq1 2106 . . . 4  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
65orbi1d 746 . . 3  |-  ( x  =  (/)  ->  ( ( x  =  (/)  \/  ph ) 
<->  ( (/)  =  (/)  \/  ph ) ) )
76elrab 2793 . 2  |-  ( (/)  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  <-> 
( (/)  e.  { (/) ,  { (/) } }  /\  ( (/)  =  (/)  \/  ph ) ) )
82, 4, 7mpbir2an 894 1  |-  (/)  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 670    = wceq 1299    e. wcel 1448   {crab 2379   (/)c0 3310   {csn 3474   {cpr 3475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-nul 3994
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-nul 3311  df-sn 3480  df-pr 3481
This theorem is referenced by:  onsucelsucexmidlem  4382  onsucelsucexmid  4383  acexmidlem2  5703
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