Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  intssuni2 Structured version   Visualization version   GIF version

Theorem intssuni2 4474
 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 4471 . 2 (𝐴 ≠ ∅ → 𝐴 𝐴)
2 uniss 4431 . 2 (𝐴𝐵 𝐴 𝐵)
31, 2sylan9ssr 3602 1 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ≠ wne 2790   ⊆ wss 3560  ∅c0 3897  ∪ cuni 4409  ∩ cint 4447 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-ne 2791  df-ral 2913  df-rex 2914  df-v 3192  df-dif 3563  df-in 3567  df-ss 3574  df-nul 3898  df-uni 4410  df-int 4448 This theorem is referenced by:  rintn0  4592  fival  8278  mremre  16204  submre  16205  lssintcl  18904  iundifdifd  29266  iundifdif  29267  ismrcd1  36780
 Copyright terms: Public domain W3C validator