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Theorem intssuni2 4892
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 4889 . 2 (𝐴 ≠ ∅ → 𝐴 𝐴)
2 uniss 4851 . 2 (𝐴𝐵 𝐴 𝐵)
31, 2sylan9ssr 3978 1 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wne 3013  wss 3933  c0 4288   cuni 4830   cint 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-uni 4831  df-int 4868
This theorem is referenced by:  rintn0  5021  fival  8864  mremre  16863  submre  16864  lssintcl  19665  iundifdifd  30241  iundifdif  30242  bj-ismoored2  34294  bj-ismooredr2  34296  ismrcd1  39173
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