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Theorem tpid2g 3721
Description: Closed theorem form of tpid2 3720. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
tpid2g  |-  ( A  e.  B  ->  A  e.  { C ,  A ,  D } )

Proof of Theorem tpid2g
StepHypRef Expression
1 eqid 2189 . . 3  |-  A  =  A
213mix2i 1172 . 2  |-  ( A  =  C  \/  A  =  A  \/  A  =  D )
3 eltpg 3652 . 2  |-  ( A  e.  B  ->  ( A  e.  { C ,  A ,  D }  <->  ( A  =  C  \/  A  =  A  \/  A  =  D )
) )
42, 3mpbiri 168 1  |-  ( A  e.  B  ->  A  e.  { C ,  A ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ w3o 979    = wceq 1364    e. wcel 2160   {ctp 3609
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3or 981  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-tp 3615
This theorem is referenced by:  rngplusgg  12645  srngplusgd  12656  lmodplusgd  12674  ipsaddgd  12686  ipsvscad  12689  topgrpplusgd  12706
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