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

Theorem difpr 4761
Description: Removing two elements as pair of elements corresponds to removing each of the two elements as singletons. (Contributed by Alexander van der Vekens, 13-Jul-2018.)
Assertion
Ref Expression
difpr (𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶})

Proof of Theorem difpr
StepHypRef Expression
1 df-pr 4587 . . 3 {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
21difeq2i 4077 . 2 (𝐴 ∖ {𝐵, 𝐶}) = (𝐴 ∖ ({𝐵} ∪ {𝐶}))
3 difun1 4247 . 2 (𝐴 ∖ ({𝐵} ∪ {𝐶})) = ((𝐴 ∖ {𝐵}) ∖ {𝐶})
42, 3eqtri 2764 1 (𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  cdif 3905  cun 3906  {csn 4584  {cpr 4586
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-pr 4587
This theorem is referenced by:  hashdifpr  14307  nbgrssvwo2  28196  nbupgrres  28198  nbupgruvtxres  28241  uvtxupgrres  28242  pmtrcnelor  31825
  Copyright terms: Public domain W3C validator