Theorem disjiunel 31066

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 3908	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
4	3	snssd 4753	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝐴)
5	1, 4	unssd 4130	. . . 4 ⊢ (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴)
6		disjiunel.1	. . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
7		disjss1 5057	. . . 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 3909	. . . 4 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐸)
10		disjiunel.2	. . . . 5 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)
11	10	disjunsn 31064	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐸) → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)))
12	3, 9, 11	syl2anc 584	. . 3 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)))
13	8, 12	mpbid 231	. 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅))
14	13	simprd 496	1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105   ∖ cdif 3893   ∪ cun 3894   ∩ cin 3895   ⊆ wss 3896  ∅c0 4266  {csn 4570  ∪ ciun 4936  Disj wdisj 5051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rmo 3349  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-sn 4571  df-iun 4938  df-disj 5052

This theorem is referenced by:  disjuniel  31067