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Theorem disjiun2 39540
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 4634 . . . 4 (𝐷 ∈ (𝐴𝐶) → 𝑥 ∈ {𝐷}𝐵 = 𝐸)
41, 3syl 17 . . 3 (𝜑 𝑥 ∈ {𝐷}𝐵 = 𝐸)
54ineq2d 3847 . 2 (𝜑 → ( 𝑥𝐶 𝐵 𝑥 ∈ {𝐷}𝐵) = ( 𝑥𝐶 𝐵𝐸))
6 disjiun2.1 . . 3 (𝜑Disj 𝑥𝐴 𝐵)
7 disjiun2.2 . . 3 (𝜑𝐶𝐴)
8 eldifi 3765 . . . 4 (𝐷 ∈ (𝐴𝐶) → 𝐷𝐴)
9 snssi 4371 . . . 4 (𝐷𝐴 → {𝐷} ⊆ 𝐴)
101, 8, 93syl 18 . . 3 (𝜑 → {𝐷} ⊆ 𝐴)
111eldifbd 3620 . . . 4 (𝜑 → ¬ 𝐷𝐶)
12 disjsn 4278 . . . 4 ((𝐶 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷𝐶)
1311, 12sylibr 224 . . 3 (𝜑 → (𝐶 ∩ {𝐷}) = ∅)
14 disjiun 4672 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝐶𝐴 ∧ {𝐷} ⊆ 𝐴 ∧ (𝐶 ∩ {𝐷}) = ∅)) → ( 𝑥𝐶 𝐵 𝑥 ∈ {𝐷}𝐵) = ∅)
156, 7, 10, 13, 14syl13anc 1368 . 2 (𝜑 → ( 𝑥𝐶 𝐵 𝑥 ∈ {𝐷}𝐵) = ∅)
165, 15eqtr3d 2687 1 (𝜑 → ( 𝑥𝐶 𝐵𝐸) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wcel 2030  cdif 3604  cin 3606  wss 3607  c0 3948  {csn 4210   ciun 4552  Disj wdisj 4652
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-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rmo 2949  df-v 3233  df-sbc 3469  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-iun 4554  df-disj 4653
This theorem is referenced by:  caratheodorylem1  41061
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