Theorem tpid1g 3688

Description: Closed theorem form of tpid1 3687. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Assertion

Ref	Expression
tpid1g	|- ( A e. B -> A e. { A , C , D } )

Proof of Theorem tpid1g

Step	Hyp	Ref	Expression
1		eqid 2165	. . 3 |- A = A
2	1	3mix1i 1159	. 2 |- ( A = A \/ A = C \/ A = D )
3		eltpg 3621	. 2 |- ( A e. B -> ( A e. { A , C , D } <-> ( A = A \/ A = C \/ A = D ) ) )
4	2, 3	mpbiri 167	1 |- ( A e. B -> A e. { A , C , D } )

Syntax hints:   -> wi 4   \/ w3o 967   = wceq 1343   e. wcel 2136   {ctp 3578

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3or 969  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-tp 3584

This theorem is referenced by:  rngbaseg  12511  srngbased  12518  lmodbased  12529  ipsbased  12537  ipsscad  12540  topgrpbasd  12547