Theorem tpid1g 3744

Description: Closed theorem form of tpid1 3743. (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 2204	. . 3 |- A = A
2	1	3mix1i 1171	. 2 |- ( A = A \/ A = C \/ A = D )
3		eltpg 3677	. 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 979   = wceq 1372   e. wcel 2175   {ctp 3634

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3or 981  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-tp 3640

This theorem is referenced by:  rngbaseg  12910  srngbased  12921  lmodbased  12939  ipsbased  12951  ipsscad  12954  topgrpbasd  12971  psrbasg  14378