Theorem tpid2g 3637

Description: Closed theorem form of tpid2 3636. (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 2139	. . 3 |- A = A
2	1	3mix2i 1154	. 2 |- ( A = C \/ A = A \/ A = D )
3		eltpg 3569	. 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 961   = wceq 1331   e. wcel 1480   {ctp 3529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3or 963  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-tp 3535

This theorem is referenced by:  rngplusgg  12076  srngplusgd  12083  lmodplusgd  12094  ipsaddgd  12102  ipsvscad  12105  topgrpplusgd  12112