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Theorem ssel 3267
 Description: Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ssel (A B → (C AC B))

Proof of Theorem ssel
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 dfss2 3262 . . . . . 6 (A Bx(x Ax B))
21biimpi 186 . . . . 5 (A Bx(x Ax B))
3219.21bi 1758 . . . 4 (A B → (x Ax B))
43anim2d 548 . . 3 (A B → ((x = C x A) → (x = C x B)))
54eximdv 1622 . 2 (A B → (x(x = C x A) → x(x = C x B)))
6 df-clel 2349 . 2 (C Ax(x = C x A))
7 df-clel 2349 . 2 (C Bx(x = C x B))
85, 6, 73imtr4g 261 1 (A B → (C AC B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  ssel2  3268  sseli  3269  sseld  3272  sstr2  3279  ssralv  3330  ssrexv  3331  ralss  3332  rexss  3333  ssconb  3399  sscon  3400  ssdif  3401  unss1  3432  ssrin  3480  difin2  3516  reuss2  3535  reupick  3539  sssn  3864  uniss  3912  ss2iun  3984  ssiun  4008  iinss  4017  ssofss  4076  unipw  4117  sspwb  4118  pwadjoin  4119  eqpw1uni  4330  ssopab2b  4713  ssrel  4844  xpss12  4855  cnvss  4885  dmss  4906  dmcosseq  4973  ssreseq  4997  iss  5000  resopab2  5001  ssrnres  5059  dfco2a  5081  cores  5084  funssres  5144  fununi  5160  funimass3  5404  funfvima2  5460  f1elima  5474  resoprab2  5582  clos1conn  5879
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