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Theorem ssini 4205
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 471 . 2 (𝐴𝐵𝐴𝐶)
4 ssin 4204 . 2 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
53, 4mpbi 231 1 𝐴 ⊆ (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  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-v 3494  df-in 3940  df-ss 3949
This theorem is referenced by:  inv1  4345  cnvrescnv  6045  hartogslem1  8994  xptrrel  14328  fbasrn  22420  limciun  24419  hlimcaui  28940  chdmm1i  29181  chm0i  29194  ledii  29240  lejdii  29242  mayetes3i  29433  mdslj2i  30024  mdslmd2i  30034  sumdmdlem2  30123  sigapildsys  31320  ssoninhaus  33693  bj-disj2r  34237  bj-idres  34344  bj-rvecsscvec  34473  icomnfinre  41704  fouriersw  42393  sge0split  42568
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