Theorem tpid1g 3667

Description: Closed theorem form of tpid1 3666. (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 2154	. . 3 |- A = A
2	1	3mix1i 1154	. 2 |- ( A = A \/ A = C \/ A = D )
3		eltpg 3600	. 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 962   = wceq 1332   e. wcel 2125   {ctp 3558

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-3or 964  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-un 3102  df-sn 3562  df-pr 3563  df-tp 3564

This theorem is referenced by:  rngbaseg  12245  srngbased  12252  lmodbased  12263  ipsbased  12271  ipsscad  12274  topgrpbasd  12281