Theorem disjiun2 45836

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 5058	. . . 4 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → ∪ 𝑥 ∈ {𝐷}𝐵 = 𝐸)
4	1, 3	syl 18	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ {𝐷}𝐵 = 𝐸)
5	4	ineq2d 4173	. 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 4085	. . . 4 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ 𝐴)
9		snssi 4753	. . . 4 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴)
10	1, 8, 9	3syl 19	. . 3 ⊢ (𝜑 → {𝐷} ⊆ 𝐴)
11	1	eldifbd 3919	. . . 4 ⊢ (𝜑 → ¬ 𝐷 ∈ 𝐶)
12		disjsn 4679	. . . 4 ⊢ ((𝐶 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷 ∈ 𝐶)
13	11, 12	sylibr 237	. . 3 ⊢ (𝜑 → (𝐶 ∩ {𝐷}) = ∅)
14		disjiun 5099	. . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ {𝐷} ⊆ 𝐴 ∧ (𝐶 ∩ {𝐷}) = ∅)) → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = ∅)
15	6, 7, 10, 13, 14	syl13anc 1399	. 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = ∅)
16	5, 15	eqtr3d 2802	1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2146   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {csn 4591  ∪ ciun 4958  Disj wdisj 5078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-rab 3419  df-v 3459  df-dif 3909  df-in 3913  df-ss 3923  df-nul 4287  df-sn 4592  df-iun 4960  df-disj 5079

This theorem is used by:  caratheodorylem1  47298