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Theorem disjtpsn 4283
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 4215 . . 3 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
21ineq1i 3843 . 2 ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷})
3 disjprsn 4282 . . . . 5 ((𝐴𝐷𝐵𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
433adant3 1101 . . . 4 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
5 disjsn2 4279 . . . . 5 (𝐶𝐷 → ({𝐶} ∩ {𝐷}) = ∅)
653ad2ant3 1104 . . . 4 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐶} ∩ {𝐷}) = ∅)
74, 6jca 553 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (({𝐴, 𝐵} ∩ {𝐷}) = ∅ ∧ ({𝐶} ∩ {𝐷}) = ∅))
8 undisj1 4062 . . 3 ((({𝐴, 𝐵} ∩ {𝐷}) = ∅ ∧ ({𝐶} ∩ {𝐷}) = ∅) ↔ (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) = ∅)
97, 8sylib 208 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) = ∅)
102, 9syl5eq 2697 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wne 2823  cun 3605  cin 3606  c0 3948  {csn 4210  {cpr 4212  {ctp 4214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213  df-tp 4215
This theorem is referenced by:  disjtp2  4284  cnfldfun  19806
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