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Theorem intssuni2 4694
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 4691 . 2 (𝐴 ≠ ∅ → 𝐴 𝐴)
2 uniss 4653 . 2 (𝐴𝐵 𝐴 𝐵)
31, 2sylan9ssr 3812 1 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wne 2978  wss 3769  c0 4116   cuni 4630   cint 4669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-v 3393  df-dif 3772  df-in 3776  df-ss 3783  df-nul 4117  df-uni 4631  df-int 4670
This theorem is referenced by:  rintn0  4811  fival  8553  mremre  16465  submre  16466  lssintcl  19167  iundifdifd  29701  iundifdif  29702  bj-ismoored2  33369  bj-ismooredr2  33371  ismrcd1  37757
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