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Theorem acexmidlemb 5869
Description: Lemma for acexmid 5876. (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
acexmidlemb  |-  ( (/)  e.  B  ->  ph )
Distinct variable groups:    x, A    x, B    x, C    ph, x

Proof of Theorem acexmidlemb
StepHypRef Expression
1 acexmidlem.b . . . 4  |-  B  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }
21eleq2i 2244 . . 3  |-  ( (/)  e.  B  <->  (/)  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) } )
3 0ex 4132 . . . . 5  |-  (/)  e.  _V
43prid1 3700 . . . 4  |-  (/)  e.  { (/)
,  { (/) } }
5 eqeq1 2184 . . . . . 6  |-  ( x  =  (/)  ->  ( x  =  { (/) }  <->  (/)  =  { (/)
} ) )
65orbi1d 791 . . . . 5  |-  ( x  =  (/)  ->  ( ( x  =  { (/) }  \/  ph )  <->  ( (/)  =  { (/)
}  \/  ph )
) )
76elrab3 2896 . . . 4  |-  ( (/)  e.  { (/) ,  { (/) } }  ->  ( (/)  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  { (/)
}  \/  ph ) } 
<->  ( (/)  =  { (/)
}  \/  ph )
) )
84, 7ax-mp 5 . . 3  |-  ( (/)  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }  <->  ( (/)  =  { (/)
}  \/  ph )
)
92, 8bitri 184 . 2  |-  ( (/)  e.  B  <->  ( (/)  =  { (/)
}  \/  ph )
)
10 noel 3428 . . . 4  |-  -.  (/)  e.  (/)
113snid 3625 . . . . 5  |-  (/)  e.  { (/)
}
12 eleq2 2241 . . . . 5  |-  ( (/)  =  { (/) }  ->  ( (/) 
e.  (/)  <->  (/)  e.  { (/) } ) )
1311, 12mpbiri 168 . . . 4  |-  ( (/)  =  { (/) }  ->  (/)  e.  (/) )
1410, 13mto 662 . . 3  |-  -.  (/)  =  { (/)
}
15 orel1 725 . . 3  |-  ( -.  (/)  =  { (/) }  ->  ( ( (/)  =  { (/)
}  \/  ph )  ->  ph ) )
1614, 15ax-mp 5 . 2  |-  ( (
(/)  =  { (/) }  \/  ph )  ->  ph )
179, 16sylbi 121 1  |-  ( (/)  e.  B  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 105    \/ wo 708    = wceq 1353    e. wcel 2148   {crab 2459   (/)c0 3424   {csn 3594   {cpr 3595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4131
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-nul 3425  df-sn 3600  df-pr 3601
This theorem is referenced by:  acexmidlem1  5873
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