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Theorem inss 3804
 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 3803 . 2 (𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
2 incom 3767 . . 3 (𝐴𝐵) = (𝐵𝐴)
3 ssinss1 3803 . . 3 (𝐵𝐶 → (𝐵𝐴) ⊆ 𝐶)
42, 3syl5eqss 3612 . 2 (𝐵𝐶 → (𝐴𝐵) ⊆ 𝐶)
51, 4jaoi 393 1 ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∩ 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:  pmatcoe1fsupp  20325  ppttop  20621  inindif  28738  iunrelexp0  37013  ntrclsk3  37388  icccncfext  38773
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