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Theorem difpr 4470
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 4320 . . 3 {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
21difeq2i 3876 . 2 (𝐴 ∖ {𝐵, 𝐶}) = (𝐴 ∖ ({𝐵} ∪ {𝐶}))
3 difun1 4036 . 2 (𝐴 ∖ ({𝐵} ∪ {𝐶})) = ((𝐴 ∖ {𝐵}) ∖ {𝐶})
42, 3eqtri 2793 1 (𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cdif 3720  cun 3721  {csn 4317  {cpr 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-pr 4320
This theorem is referenced by:  hashdifpr  13405  nbgrssvwo2  26481  nbgrssvwo2OLD  26484  nbupgrres  26488  nbupgruvtxres  26537  uvtxupgrres  26538
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