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Theorem inss 3991
 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 3990 . 2 (𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
2 incom 3956 . . 3 (𝐴𝐵) = (𝐵𝐴)
3 ssinss1 3990 . . 3 (𝐵𝐶 → (𝐵𝐴) ⊆ 𝐶)
42, 3syl5eqss 3798 . 2 (𝐵𝐶 → (𝐴𝐵) ⊆ 𝐶)
51, 4jaoi 846 1 ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 836   ∩ cin 3722   ⊆ wss 3723 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-ss 3737 This theorem is referenced by:  pmatcoe1fsupp  20726  ppttop  21032  inindif  29691  iunrelexp0  38520  ntrclsk3  38894  icccncfext  40615
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