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Theorem intssuni2 4876
 Description: Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)
Assertion
Ref Expression
intssuni2 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 𝐵)

Proof of Theorem intssuni2
StepHypRef Expression
1 intssuni 4873 . 2 (𝐴 ≠ ∅ → 𝐴 𝐴)
2 uniss 4821 . 2 (𝐴𝐵 𝐴 𝐵)
31, 2sylan9ssr 3956 1 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ≠ wne 3011   ⊆ wss 3908  ∅c0 4265  ∪ cuni 4813  ∩ cint 4851 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-in 3915  df-ss 3925  df-nul 4266  df-uni 4814  df-int 4852 This theorem is referenced by:  rintn0  5006  fival  8864  mremre  16866  submre  16867  lssintcl  19727  iundifdifd  30320  iundifdif  30321  bj-ismoored2  34484  bj-ismooredr2  34486  ismrcd1  39569
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