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Theorem ss2in 3989
Description: Intersection of subclasses. (Contributed by NM, 5-May-2000.)
Assertion
Ref Expression
ss2in ((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))

Proof of Theorem ss2in
StepHypRef Expression
1 ssrin 3986 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 sslin 3987 . 2 (𝐶𝐷 → (𝐵𝐶) ⊆ (𝐵𝐷))
31, 2sylan9ss 3765 1 ((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  cin 3722  wss 3723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-ss 3737
This theorem is referenced by:  disjxiun  4783  undom  8204  strlemor1OLD  16177  strleun  16180  dprdss  18636  dprd2da  18649  ablfac1b  18677  tgcl  20994  innei  21150  hausnei2  21378  bwth  21434  fbssfi  21861  fbunfip  21893  fgcl  21902  blin2  22454  vtxdun  26612  vtxdginducedm1  26674  5oai  28860  mayetes3i  28928  mdsl0  29509  neibastop1  32691  ismblfin  33783  heibor1lem  33940  pl42lem2N  35788  pl42lem3N  35789  ntrk2imkb  38861  ssin0  39744
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