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Theorem ssun 3776
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 3762 . 2 (𝐴𝐵𝐴 ⊆ (𝐵𝐶))
2 ssun4 3763 . 2 (𝐴𝐶𝐴 ⊆ (𝐵𝐶))
31, 2jaoi 394 1 ((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  cun 3558  wss 3560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-un 3565  df-in 3567  df-ss 3574
This theorem is referenced by:  pwunss  4989  pwssun  4990  ordssun  5796  padct  29381
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