Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjtpsn Structured version   Visualization version   GIF version

Theorem disjtpsn 4650
 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 4571 . . 3 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
21ineq1i 4184 . 2 ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷})
3 disjprsn 4649 . . . . 5 ((𝐴𝐷𝐵𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
433adant3 1128 . . . 4 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
5 disjsn2 4647 . . . . 5 (𝐶𝐷 → ({𝐶} ∩ {𝐷}) = ∅)
653ad2ant3 1131 . . . 4 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐶} ∩ {𝐷}) = ∅)
74, 6jca 514 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (({𝐴, 𝐵} ∩ {𝐷}) = ∅ ∧ ({𝐶} ∩ {𝐷}) = ∅))
8 undisj1 4410 . . 3 ((({𝐴, 𝐵} ∩ {𝐷}) = ∅ ∧ ({𝐶} ∩ {𝐷}) = ∅) ↔ (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) = ∅)
97, 8sylib 220 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) = ∅)
102, 9syl5eq 2868 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ≠ wne 3016   ∪ cun 3933   ∩ cin 3934  ∅c0 4290  {csn 4566  {cpr 4568  {ctp 4570 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-sn 4567  df-pr 4569  df-tp 4571 This theorem is referenced by:  disjtp2  4651  cnfldfun  20556
 Copyright terms: Public domain W3C validator