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Theorem acexmidlema 5625
Description: Lemma for acexmid 5633. (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 2154 . . 3  |-  ( {
(/) }  e.  A  <->  {
(/) }  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
3 p0ex 4014 . . . . 5  |-  { (/) }  e.  _V
43prid2 3544 . . . 4  |-  { (/) }  e.  { (/) ,  { (/)
} }
5 eqeq1 2094 . . . . . 6  |-  ( x  =  { (/) }  ->  ( x  =  (/)  <->  { (/) }  =  (/) ) )
65orbi1d 740 . . . . 5  |-  ( x  =  { (/) }  ->  ( ( x  =  (/)  \/ 
ph )  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
76elrab3 2770 . . . 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 3288 . . . 4  |-  -.  (/)  e.  (/)
11 0ex 3958 . . . . . 6  |-  (/)  e.  _V
1211snid 3470 . . . . 5  |-  (/)  e.  { (/)
}
13 eleq2 2151 . . . . 5  |-  ( {
(/) }  =  (/)  ->  ( (/) 
e.  { (/) }  <->  (/)  e.  (/) ) )
1412, 13mpbii 146 . . . 4  |-  ( {
(/) }  =  (/)  ->  (/)  e.  (/) )
1510, 14mto 623 . . 3  |-  -.  { (/)
}  =  (/)
16 orel1 679 . . 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 664    = wceq 1289    e. wcel 1438   {crab 2363   (/)c0 3284   {csn 3441   {cpr 3442
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448
This theorem is referenced by:  acexmidlem1  5630
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