Theorem tpid1g 3784

Description: Closed theorem form of tpid1 3783. (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 1195	. 2 |- ( A = A \/ A = C \/ A = D )
3		eltpg 3714	. 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 1003   = wceq 1397   e. wcel 2202   {ctp 3671

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-tp 3677

This theorem is referenced by:  rngbaseg  13218  srngbased  13229  lmodbased  13247  ipsbased  13259  ipsscad  13262  topgrpbasd  13279  psrbasg  14687