Theorem disjiunel 32796

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 3916	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
4	3	snssd 4745	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝐴)
5	1, 4	unssd 4144	. . . 4 ⊢ (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴)
6		disjiunel.1	. . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
7		disjss1 5073	. . . 4 ⊢ ((𝐸 ∪ {𝑌}) ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵))
8	5, 6, 7	sylc 65	. . 3 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵)
9	2	eldifbd 3917	. . . 4 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐸)
10		disjiunel.2	. . . . 5 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)
11	10	disjunsn 32794	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐸) → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)))
12	3, 9, 11	syl2anc 593	. . 3 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)))
13	8, 12	mpbid 234	. 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅))
14	13	simprd 499	1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ∖ cdif 3901   ∪ cun 3902   ∩ 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-nfc 2911  df-ral 3077  df-rex 3087  df-rmo 3367  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-sn 4583  df-iun 4951  df-disj 5068

This theorem is referenced by:  disjuniel  32797