Theorem tpid2g 3811

Description: Closed theorem form of tpid2 3810. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Assertion

Ref	Expression
tpid2g	|- ( A e. B -> A e. { C , A , D } )

Proof of Theorem tpid2g

Step	Hyp	Ref	Expression
1		eqid 2234	. . 3 |- A = A
2	1	3mix2i 1197	. 2 |- ( A = C \/ A = A \/ A = D )
3		eltpg 3739	. 2 |- ( A e. B -> ( A e. { C , A , D } <-> ( A = C \/ A = A \/ A = D ) ) )
4	2, 3	mpbiri 168	1 |- ( A e. B -> A e. { C , A , D } )

Syntax hints:   -> wi 4   \/ w3o 1004   = wceq 1398   e. wcel 2205   {ctp 3696

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-tp 3702

This theorem is referenced by:  rngplusgg  13434  srngplusgd  13445  lmodplusgd  13463  ipsaddgd  13475  ipsvscad  13478  topgrpplusgd  13495