Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjiun2 Structured version   Visualization version   GIF version

Theorem disjiun2 42279
Description: In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
disjiun2.1 (𝜑Disj 𝑥𝐴 𝐵)
disjiun2.2 (𝜑𝐶𝐴)
disjiun2.3 (𝜑𝐷 ∈ (𝐴𝐶))
disjiun2.4 (𝑥 = 𝐷𝐵 = 𝐸)
Assertion
Ref Expression
disjiun2 (𝜑 → ( 𝑥𝐶 𝐵𝐸) = ∅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝐸
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem disjiun2
StepHypRef Expression
1 disjiun2.3 . . . 4 (𝜑𝐷 ∈ (𝐴𝐶))
2 disjiun2.4 . . . . 5 (𝑥 = 𝐷𝐵 = 𝐸)
32iunxsng 4998 . . . 4 (𝐷 ∈ (𝐴𝐶) → 𝑥 ∈ {𝐷}𝐵 = 𝐸)
41, 3syl 17 . . 3 (𝜑 𝑥 ∈ {𝐷}𝐵 = 𝐸)
54ineq2d 4127 . 2 (𝜑 → ( 𝑥𝐶 𝐵 𝑥 ∈ {𝐷}𝐵) = ( 𝑥𝐶 𝐵𝐸))
6 disjiun2.1 . . 3 (𝜑Disj 𝑥𝐴 𝐵)
7 disjiun2.2 . . 3 (𝜑𝐶𝐴)
8 eldifi 4041 . . . 4 (𝐷 ∈ (𝐴𝐶) → 𝐷𝐴)
9 snssi 4721 . . . 4 (𝐷𝐴 → {𝐷} ⊆ 𝐴)
101, 8, 93syl 18 . . 3 (𝜑 → {𝐷} ⊆ 𝐴)
111eldifbd 3879 . . . 4 (𝜑 → ¬ 𝐷𝐶)
12 disjsn 4627 . . . 4 ((𝐶 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷𝐶)
1311, 12sylibr 237 . . 3 (𝜑 → (𝐶 ∩ {𝐷}) = ∅)
14 disjiun 5040 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝐶𝐴 ∧ {𝐷} ⊆ 𝐴 ∧ (𝐶 ∩ {𝐷}) = ∅)) → ( 𝑥𝐶 𝐵 𝑥 ∈ {𝐷}𝐵) = ∅)
156, 7, 10, 13, 14syl13anc 1374 . 2 (𝜑 → ( 𝑥𝐶 𝐵 𝑥 ∈ {𝐷}𝐵) = ∅)
165, 15eqtr3d 2779 1 (𝜑 → ( 𝑥𝐶 𝐵𝐸) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1543  wcel 2110  cdif 3863  cin 3865  wss 3866  c0 4237  {csn 4541   ciun 4904  Disj wdisj 5018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rmo 3069  df-rab 3070  df-v 3410  df-dif 3869  df-in 3873  df-ss 3883  df-nul 4238  df-sn 4542  df-iun 4906  df-disj 5019
This theorem is referenced by:  caratheodorylem1  43739
  Copyright terms: Public domain W3C validator