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Theorem sucdifsn2 38219
Description: Absorption of union with a singleton by difference. (Contributed by Peter Mazsa, 24-Jul-2024.)
Assertion
Ref Expression
sucdifsn2 ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴

Proof of Theorem sucdifsn2
StepHypRef Expression
1 disjcsn 9642 . 2 (𝐴 ∩ {𝐴}) = ∅
2 undif5 4491 . 2 ((𝐴 ∩ {𝐴}) = ∅ → ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴)
31, 2ax-mp 5 1 ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cdif 3960  cun 3961  cin 3962  c0 4339  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-nul 4340  df-sn 4632  df-pr 4634
This theorem is referenced by:  sucdifsn  38220  partsuc2  38761
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