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Theorem onsucelsucexmidlem1 4413
Description: Lemma for onsucelsucexmid 4415. (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 4025 . . 3  |-  (/)  e.  _V
21prid1 3599 . 2  |-  (/)  e.  { (/)
,  { (/) } }
3 eqid 2117 . . 3  |-  (/)  =  (/)
43orci 705 . 2  |-  ( (/)  =  (/)  \/  ph )
5 eqeq1 2124 . . . 4  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
65orbi1d 765 . . 3  |-  ( x  =  (/)  ->  ( ( x  =  (/)  \/  ph ) 
<->  ( (/)  =  (/)  \/  ph ) ) )
76elrab 2813 . 2  |-  ( (/)  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  <-> 
( (/)  e.  { (/) ,  { (/) } }  /\  ( (/)  =  (/)  \/  ph ) ) )
82, 4, 7mpbir2an 911 1  |-  (/)  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 682    = wceq 1316    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-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024
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-nul 3334  df-sn 3503  df-pr 3504
This theorem is referenced by:  onsucelsucexmidlem  4414  onsucelsucexmid  4415  acexmidlem2  5739
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