Theorem disjiunel 30345

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 3947	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
4	3	snssd 4741	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝐴)
5	1, 4	unssd 4161	. . . 4 ⊢ (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴)
6		disjiunel.1	. . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
7		disjss1 5036	. . . 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 3948	. . . 4 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐸)
10		disjiunel.2	. . . . 5 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)
11	10	disjunsn 30343	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐸) → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)))
12	3, 9, 11	syl2anc 586	. . 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 498	1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ∖ cdif 3932   ∪ cun 3933   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  {csn 4566  ∪ ciun 4918  Disj wdisj 5030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-sn 4567  df-iun 4920  df-disj 5031

This theorem is referenced by:  disjuniel  30346