Theorem difpr 4771

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

Step	Hyp	Ref	Expression
1		df-pr 4592	. . 3 ⊢ {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
2	1	difeq2i 4078	. 2 ⊢ (𝐴 ∖ {𝐵, 𝐶}) = (𝐴 ∖ ({𝐵} ∪ {𝐶}))
3		difun1 4252	. 2 ⊢ (𝐴 ∖ ({𝐵} ∪ {𝐶})) = ((𝐴 ∖ {𝐵}) ∖ {𝐶})
4	2, 3	eqtri 2786	1 ⊢ (𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶})

Syntax hints:   = wceq 1570   ∖ cdif 3902   ∪ cun 3903  {csn 4589  {cpr 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-pr 4592

This theorem is referenced by:  hashdifpr  14448  chnccat  18677  nbgrssvwo2  29712  nbupgrres  29714  nbupgruvtxres  29757  uvtxupgrres  29758  pmtrcnelor  33411