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Theorem inss 4038
Description: Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
Assertion
Ref Expression
inss ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)

Proof of Theorem inss
StepHypRef Expression
1 ssinss1 4037 . 2 (𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
2 incom 4003 . . 3 (𝐴𝐵) = (𝐵𝐴)
3 ssinss1 4037 . . 3 (𝐵𝐶 → (𝐵𝐴) ⊆ 𝐶)
42, 3syl5eqss 3845 . 2 (𝐵𝐶 → (𝐴𝐵) ⊆ 𝐶)
51, 4jaoi 884 1 ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 874  cin 3768  wss 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-in 3776  df-ss 3783
This theorem is referenced by:  pmatcoe1fsupp  20834  ppttop  21140  inindif  29871  iunrelexp0  38777  ntrclsk3  39150  icccncfext  40844
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