Theorem tpid1g 3788

Description: Closed theorem form of tpid1 3787. (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 2231	. . 3 |- A = A
2	1	3mix1i 1196	. 2 |- ( A = A \/ A = C \/ A = D )
3		eltpg 3718	. 2 |- ( A e. B -> ( A e. { A , C , D } <-> ( A = A \/ A = C \/ A = D ) ) )
4	2, 3	mpbiri 168	1 |- ( A e. B -> A e. { A , C , D } )

Syntax hints:   -> wi 4   \/ w3o 1004   = wceq 1398   e. wcel 2202   {ctp 3675

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-tp 3681

This theorem is referenced by:  rngbaseg  13299  srngbased  13310  lmodbased  13328  ipsbased  13340  ipsscad  13343  topgrpbasd  13360  psrbasg  14775