Theorem tpid2g 3825

Description: Closed theorem form of tpid2 3824. (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 2238	. . 3 |- A = A
2	1	3mix2i 1201	. 2 |- ( A = C \/ A = A \/ A = D )
3		eltpg 3753	. 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 1008   = wceq 1402   e. wcel 2209   {ctp 3710

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-tp 3716

This theorem is referenced by:  rngplusgg  13474  srngplusgd  13485  lmodplusgd  13503  ipsaddgd  13515  ipsvscad  13518  topgrpplusgd  13535