Theorem disjiun2 45669

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 5060	. . . 4 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → ∪ 𝑥 ∈ {𝐷}𝐵 = 𝐸)
4	1, 3	syl 18	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ {𝐷}𝐵 = 𝐸)
5	4	ineq2d 4181	. 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 4093	. . . 4 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ 𝐴)
9		snssi 4756	. . . 4 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴)
10	1, 8, 9	3syl 19	. . 3 ⊢ (𝜑 → {𝐷} ⊆ 𝐴)
11	1	eldifbd 3926	. . . 4 ⊢ (𝜑 → ¬ 𝐷 ∈ 𝐶)
12		disjsn 4682	. . . 4 ⊢ ((𝐶 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷 ∈ 𝐶)
13	11, 12	sylibr 237	. . 3 ⊢ (𝜑 → (𝐶 ∩ {𝐷}) = ∅)
14		disjiun 5101	. . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ {𝐷} ⊆ 𝐴 ∧ (𝐶 ∩ {𝐷}) = ∅)) → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = ∅)
15	6, 7, 10, 13, 14	syl13anc 1397	. 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = ∅)
16	5, 15	eqtr3d 2806	1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1567   ∈ wcel 2149   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  {csn 4594  ∪ ciun 4960  Disj wdisj 5080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-rab 3424  df-v 3465  df-dif 3916  df-in 3920  df-ss 3930  df-nul 4295  df-sn 4595  df-iun 4962  df-disj 5081

This theorem is referenced by:  caratheodorylem1  47131