Theorem disjiun2 45635

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	⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
disjiun2.2	⊢ (𝜑 → 𝐶 ⊆ 𝐴)
disjiun2.3	⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶))
disjiun2.4	⊢ (𝑥 = 𝐷 → 𝐵 = 𝐸)

Assertion

Ref	Expression
disjiun2	⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸) = ∅)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝐸

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem disjiun2

Step	Hyp	Ref	Expression
1		disjiun2.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶))
2		disjiun2.4	. . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐸)
3	2	iunxsng 5047	. . . 4 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → ∪ 𝑥 ∈ {𝐷}𝐵 = 𝐸)
4	1, 3	syl 17	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ {𝐷}𝐵 = 𝐸)
5	4	ineq2d 4172	. 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸))
6		disjiun2.1	. . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
7		disjiun2.2	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
8		eldifi 4084	. . . 4 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ 𝐴)
9		snssi 4744	. . . 4 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴)
10	1, 8, 9	3syl 18	. . 3 ⊢ (𝜑 → {𝐷} ⊆ 𝐴)
11	1	eldifbd 3917	. . . 4 ⊢ (𝜑 → ¬ 𝐷 ∈ 𝐶)
12		disjsn 4670	. . . 4 ⊢ ((𝐶 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷 ∈ 𝐶)
13	11, 12	sylibr 236	. . 3 ⊢ (𝜑 → (𝐶 ∩ {𝐷}) = ∅)
14		disjiun 5088	. . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ {𝐷} ⊆ 𝐴 ∧ (𝐶 ∩ {𝐷}) = ∅)) → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = ∅)
15	6, 7, 10, 13, 14	syl13anc 1391	. 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = ∅)
16	5, 15	eqtr3d 2799	1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1560   ∈ wcel 2142   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4582  ∪ ciun 4949  Disj wdisj 5067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-rab 3415  df-v 3456  df-dif 3907  df-in 3911  df-ss 3921  df-nul 4286  df-sn 4583  df-iun 4951  df-disj 5068

This theorem is referenced by:  caratheodorylem1  47097