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Theorem acexmidlema 5534
Description: Lemma for acexmid 5542. (Contributed by Jim Kingdon, 6-Aug-2019.)
Hypotheses
Ref Expression
acexmidlem.a  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
acexmidlem.b  |-  B  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }
acexmidlem.c  |-  C  =  { A ,  B }
Assertion
Ref Expression
acexmidlema  |-  ( {
(/) }  e.  A  ->  ph )
Distinct variable groups:    x, A    x, B    x, C    ph, x

Proof of Theorem acexmidlema
StepHypRef Expression
1 acexmidlem.a . . . 4  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
21eleq2i 2146 . . 3  |-  ( {
(/) }  e.  A  <->  {
(/) }  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
3 p0ex 3967 . . . . 5  |-  { (/) }  e.  _V
43prid2 3507 . . . 4  |-  { (/) }  e.  { (/) ,  { (/)
} }
5 eqeq1 2088 . . . . . 6  |-  ( x  =  { (/) }  ->  ( x  =  (/)  <->  { (/) }  =  (/) ) )
65orbi1d 738 . . . . 5  |-  ( x  =  { (/) }  ->  ( ( x  =  (/)  \/ 
ph )  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
76elrab3 2751 . . . 4  |-  ( {
(/) }  e.  { (/) ,  { (/) } }  ->  ( { (/) }  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
84, 7ax-mp 7 . . 3  |-  ( {
(/) }  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } 
<->  ( { (/) }  =  (/) 
\/  ph ) )
92, 8bitri 182 . 2  |-  ( {
(/) }  e.  A  <->  ( { (/) }  =  (/)  \/ 
ph ) )
10 noel 3262 . . . 4  |-  -.  (/)  e.  (/)
11 0ex 3913 . . . . . 6  |-  (/)  e.  _V
1211snid 3433 . . . . 5  |-  (/)  e.  { (/)
}
13 eleq2 2143 . . . . 5  |-  ( {
(/) }  =  (/)  ->  ( (/) 
e.  { (/) }  <->  (/)  e.  (/) ) )
1412, 13mpbii 146 . . . 4  |-  ( {
(/) }  =  (/)  ->  (/)  e.  (/) )
1510, 14mto 621 . . 3  |-  -.  { (/)
}  =  (/)
16 orel1 677 . . 3  |-  ( -. 
{ (/) }  =  (/)  ->  ( ( { (/) }  =  (/)  \/  ph )  ->  ph ) )
1715, 16ax-mp 7 . 2  |-  ( ( { (/) }  =  (/)  \/ 
ph )  ->  ph )
189, 17sylbi 119 1  |-  ( {
(/) }  e.  A  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   {crab 2353   (/)c0 3258   {csn 3406   {cpr 3407
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-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413
This theorem is referenced by:  acexmidlem1  5539
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