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Theorem ssun 3754
 Description: A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
Assertion
Ref Expression
ssun ((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))

Proof of Theorem ssun
StepHypRef Expression
1 ssun3 3740 . 2 (𝐴𝐵𝐴 ⊆ (𝐵𝐶))
2 ssun4 3741 . 2 (𝐴𝐶𝐴 ⊆ (𝐵𝐶))
31, 2jaoi 393 1 ((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∪ cun 3538   ⊆ 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-un 3545  df-in 3547  df-ss 3554 This theorem is referenced by:  pwunss  4943  pwssun  4944  ordssun  5744  padct  28885
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