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Theorem disjiun2 38251
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 4538 . . . 4 (𝐷 ∈ (𝐴𝐶) → 𝑥 ∈ {𝐷}𝐵 = 𝐸)
41, 3syl 17 . . 3 (𝜑 𝑥 ∈ {𝐷}𝐵 = 𝐸)
54ineq2d 3776 . 2 (𝜑 → ( 𝑥𝐶 𝐵 𝑥 ∈ {𝐷}𝐵) = ( 𝑥𝐶 𝐵𝐸))
6 disjiun2.1 . . 3 (𝜑Disj 𝑥𝐴 𝐵)
7 disjiun2.2 . . 3 (𝜑𝐶𝐴)
8 eldifi 3694 . . . 4 (𝐷 ∈ (𝐴𝐶) → 𝐷𝐴)
9 snssi 4280 . . . 4 (𝐷𝐴 → {𝐷} ⊆ 𝐴)
101, 8, 93syl 18 . . 3 (𝜑 → {𝐷} ⊆ 𝐴)
111eldifbd 3553 . . . 4 (𝜑 → ¬ 𝐷𝐶)
12 disjsn 4192 . . . 4 ((𝐶 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷𝐶)
1311, 12sylibr 223 . . 3 (𝜑 → (𝐶 ∩ {𝐷}) = ∅)
14 disjiun 4573 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝐶𝐴 ∧ {𝐷} ⊆ 𝐴 ∧ (𝐶 ∩ {𝐷}) = ∅)) → ( 𝑥𝐶 𝐵 𝑥 ∈ {𝐷}𝐵) = ∅)
156, 7, 10, 13, 14syl13anc 1320 . 2 (𝜑 → ( 𝑥𝐶 𝐵 𝑥 ∈ {𝐷}𝐵) = ∅)
165, 15eqtr3d 2646 1 (𝜑 → ( 𝑥𝐶 𝐵𝐸) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1475  wcel 1977  cdif 3537  cin 3539  wss 3540  c0 3874  {csn 4125   ciun 4455  Disj wdisj 4553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rmo 2904  df-v 3175  df-sbc 3403  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-iun 4457  df-disj 4554
This theorem is referenced by:  caratheodorylem1  39416
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