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Theorem ssini 3798
 Description: An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
Hypotheses
Ref Expression
ssini.1 𝐴𝐵
ssini.2 𝐴𝐶
Assertion
Ref Expression
ssini 𝐴 ⊆ (𝐵𝐶)

Proof of Theorem ssini
StepHypRef Expression
1 ssini.1 . . 3 𝐴𝐵
2 ssini.2 . . 3 𝐴𝐶
31, 2pm3.2i 470 . 2 (𝐴𝐵𝐴𝐶)
4 ssin 3797 . 2 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
53, 4mpbi 219 1 𝐴 ⊆ (𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   ∩ cin 3539   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554 This theorem is referenced by:  inv1  3922  hartogslem1  8330  xptrrel  13567  fbasrn  21498  limciun  23464  hlimcaui  27477  chdmm1i  27720  chm0i  27733  ledii  27779  lejdii  27781  mayetes3i  27972  mdslj2i  28563  mdslmd2i  28573  sumdmdlem2  28662  sigapildsys  29552  ssoninhaus  31617  icomnfinre  38626  fouriersw  39124  sge0split  39302
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