Theorem disjiunel 33241

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 4050	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
4	3	snssd 4743	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝐴)
5	1, 4	unssd 4189	. . . 4 ⊢ (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴)
6		disjiunel.1	. . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
7		disjss1 5075	. . . 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 4051	. . . 4 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐸)
10		disjiunel.2	. . . . 5 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)
11	10	disjunsn 33239	. . . 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 1569   ∈ wcel 2144   ⊆ wss 3896   ∖ cdif 4044   ∩ cin 4088   ∪ cun 4148  ∅c0 4277  {csn 4584  ∪ ciun 4950  Disj wdisj 5069

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-sn 4585  df-iun 4952  df-disj 5070

This theorem is used by:  disjuniel  33242