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Theorem ss2in 4210
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 4207 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 sslin 4208 . 2 (𝐶𝐷 → (𝐵𝐶) ⊆ (𝐵𝐷))
31, 2sylan9ss 3977 1 ((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  cin 3932  wss 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-in 3940  df-ss 3949
This theorem is referenced by:  disjxiun  5054  undom  8593  strleun  16579  dprdss  19080  dprd2da  19093  ablfac1b  19121  tgcl  21505  innei  21661  hausnei2  21889  bwth  21946  fbssfi  22373  fbunfip  22405  fgcl  22414  blin2  22966  vtxdun  27190  vtxdginducedm1  27252  5oai  29365  mayetes3i  29433  mdsl0  30014  neibastop1  33604  ismblfin  34814  heibor1lem  34968  pl42lem2N  36996  pl42lem3N  36997  ntrk2imkb  40265  ssin0  41194
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