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Theorem disjtpsn 4715
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 4631 . . 3 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
21ineq1i 4216 . 2 ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷})
3 disjprsn 4714 . . . . 5 ((𝐴𝐷𝐵𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
433adant3 1133 . . . 4 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
5 disjsn2 4712 . . . . 5 (𝐶𝐷 → ({𝐶} ∩ {𝐷}) = ∅)
653ad2ant3 1136 . . . 4 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐶} ∩ {𝐷}) = ∅)
74, 6jca 511 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (({𝐴, 𝐵} ∩ {𝐷}) = ∅ ∧ ({𝐶} ∩ {𝐷}) = ∅))
8 undisj1 4462 . . 3 ((({𝐴, 𝐵} ∩ {𝐷}) = ∅ ∧ ({𝐶} ∩ {𝐷}) = ∅) ↔ (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) = ∅)
97, 8sylib 218 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) = ∅)
102, 9eqtrid 2789 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wne 2940  cun 3949  cin 3950  c0 4333  {csn 4626  {cpr 4628  {ctp 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-tp 4631
This theorem is referenced by:  disjtp2  4716  hash7g  14525  cnfldfunALT  21379  cnfldfunALTOLD  21392  cnfldfunALTOLDOLD  21393
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