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Theorem disjtpsn 4408
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 4341 . . 3 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
21ineq1i 3974 . 2 ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷})
3 disjprsn 4407 . . . . 5 ((𝐴𝐷𝐵𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
433adant3 1162 . . . 4 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
5 disjsn2 4405 . . . . 5 (𝐶𝐷 → ({𝐶} ∩ {𝐷}) = ∅)
653ad2ant3 1165 . . . 4 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐶} ∩ {𝐷}) = ∅)
74, 6jca 507 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (({𝐴, 𝐵} ∩ {𝐷}) = ∅ ∧ ({𝐶} ∩ {𝐷}) = ∅))
8 undisj1 4192 . . 3 ((({𝐴, 𝐵} ∩ {𝐷}) = ∅ ∧ ({𝐶} ∩ {𝐷}) = ∅) ↔ (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) = ∅)
97, 8sylib 209 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) = ∅)
102, 9syl5eq 2811 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wne 2937  cun 3732  cin 3733  c0 4081  {csn 4336  {cpr 4338  {ctp 4340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-sn 4337  df-pr 4339  df-tp 4341
This theorem is referenced by:  disjtp2  4409  cnfldfun  20045
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