Theorem tpid2g 3584

Description: Closed theorem form of tpid2 3583. (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 2100	. . 3 |- A = A
2	1	3mix2i 1122	. 2 |- ( A = C \/ A = A \/ A = D )
3		eltpg 3516	. 2 |- ( A e. B -> ( A e. { C , A , D } <-> ( A = C \/ A = A \/ A = D ) ) )
4	2, 3	mpbiri 167	1 |- ( A e. B -> A e. { C , A , D } )

Syntax hints:   -> wi 4   \/ w3o 929   = wceq 1299   e. wcel 1448   {ctp 3476

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3or 931  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-tp 3482

This theorem is referenced by:  rngplusgg  11858  srngplusgd  11865  lmodplusgd  11876  ipsaddgd  11884  ipsvscad  11887  topgrpplusgd  11894