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Theorem ss2in 3983
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 3981 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 sslin 3982 . 2 (𝐶𝐷 → (𝐵𝐶) ⊆ (𝐵𝐷))
31, 2sylan9ss 3757 1 ((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  cin 3714  wss 3715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729
This theorem is referenced by:  disjxiun  4801  undom  8215  strlemor1OLD  16191  strleun  16194  dprdss  18648  dprd2da  18661  ablfac1b  18689  tgcl  20995  innei  21151  hausnei2  21379  bwth  21435  fbssfi  21862  fbunfip  21894  fgcl  21903  blin2  22455  vtxdun  26608  vtxdginducedm1  26670  5oai  28850  mayetes3i  28918  mdsl0  29499  neibastop1  32681  ismblfin  33781  heibor1lem  33939  pl42lem2N  35787  pl42lem3N  35788  ntrk2imkb  38855  ssin0  39740
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