Theorem disjiunel 33288

Description: A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.)

Hypotheses

Ref	Expression
disjiunel.1	⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
disjiunel.2	⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)
disjiunel.3	⊢ (𝜑 → 𝐸 ⊆ 𝐴)
disjiunel.4	⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 𝐸))

Assertion

Ref	Expression
disjiunel	⊢ (𝜑 → (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)

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

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

Proof of Theorem disjiunel

Step	Hyp	Ref	Expression
1		disjiunel.3	. . . . 5 ⊢ (𝜑 → 𝐸 ⊆ 𝐴)
2		disjiunel.4	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 𝐸))
3	2	eldifad 4054	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
4	3	snssd 4749	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝐴)
5	1, 4	unssd 4193	. . . 4 ⊢ (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴)
6		disjiunel.1	. . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
7		disjss1 5081	. . . 4 ⊢ ((𝐸 ∪ {𝑌}) ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵))
8	5, 6, 7	sylc 66	. . 3 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵)
9	2	eldifbd 4055	. . . 4 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐸)
10		disjiunel.2	. . . . 5 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)
11	10	disjunsn 33286	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐸) → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)))
12	3, 9, 11	syl2anc 596	. . 3 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)))
13	8, 12	mpbid 235	. 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅))
14	13	simprd 501	1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588  ∪ ciun 4956  Disj wdisj 5075

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

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 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-sn 4589  df-iun 4958  df-disj 5076

This theorem is used by:  disjuniel  33289