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Theorem disjtpsn 4720
Description: The disjoint intersection of an unordered triple and a singleton. (Contributed by AV, 14-Nov-2021.)
Assertion
Ref Expression
disjtpsn ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = ∅)

Proof of Theorem disjtpsn
StepHypRef Expression
1 df-tp 4636 . . 3 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
21ineq1i 4224 . 2 ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷})
3 disjprsn 4719 . . . . 5 ((𝐴𝐷𝐵𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
433adant3 1131 . . . 4 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
5 disjsn2 4717 . . . . 5 (𝐶𝐷 → ({𝐶} ∩ {𝐷}) = ∅)
653ad2ant3 1134 . . . 4 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐶} ∩ {𝐷}) = ∅)
74, 6jca 511 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (({𝐴, 𝐵} ∩ {𝐷}) = ∅ ∧ ({𝐶} ∩ {𝐷}) = ∅))
8 undisj1 4468 . . 3 ((({𝐴, 𝐵} ∩ {𝐷}) = ∅ ∧ ({𝐶} ∩ {𝐷}) = ∅) ↔ (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) = ∅)
97, 8sylib 218 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) = ∅)
102, 9eqtrid 2787 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wne 2938  cun 3961  cin 3962  c0 4339  {csn 4631  {cpr 4633  {ctp 4635
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-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-tp 4636
This theorem is referenced by:  disjtp2  4721  hash7g  14522  cnfldfunALT  21397  cnfldfunALTOLD  21410  cnfldfunALTOLDOLD  21411
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