Theorem tpid2g 3749

Description: Closed theorem form of tpid2 3748. (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 2206	. . 3 |- A = A
2	1	3mix2i 1173	. 2 |- ( A = C \/ A = A \/ A = D )
3		eltpg 3680	. 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 980   = wceq 1373   e. wcel 2177   {ctp 3637

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3or 982  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3172  df-sn 3641  df-pr 3642  df-tp 3643

This theorem is referenced by:  rngplusgg  13019  srngplusgd  13030  lmodplusgd  13048  ipsaddgd  13060  ipsvscad  13063  topgrpplusgd  13080