Theorem disjun0 33237

Description: Adding the empty element preserves disjointness. (Contributed by Thierry Arnoux, 30-May-2020.)

Assertion

Ref	Expression
disjun0	⊢ (Disj 𝑥 ∈ 𝐴 𝑥 → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem disjun0

Step	Hyp	Ref	Expression
1		snssi 4742	. . . . 5 ⊢ (∅ ∈ 𝐴 → {∅} ⊆ 𝐴)
2		ssequn2 4187	. . . . 5 ⊢ ({∅} ⊆ 𝐴 ↔ (𝐴 ∪ {∅}) = 𝐴)
3	1, 2	sylib 221	. . . 4 ⊢ (∅ ∈ 𝐴 → (𝐴 ∪ {∅}) = 𝐴)
4	3	disjeq1d 5077	. . 3 ⊢ (∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ Disj 𝑥 ∈ 𝐴 𝑥))
5	4	biimparc 485	. 2 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
6		simpl 488	. . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥 ∈ 𝐴 𝑥)
7		in0 4344	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅
8	7	a1i 11	. . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅)
9		0ex 5260	. . . . 5 ⊢ ∅ ∈ V
10		id 23	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
11	10	disjunsn 33236	. . . . 5 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥 ∈ 𝐴 𝑥 ∧ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅)))
12	9, 11	mpan 703	. . . 4 ⊢ (¬ ∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥 ∈ 𝐴 𝑥 ∧ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅)))
13	12	adantl 487	. . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥 ∈ 𝐴 𝑥 ∧ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅)))
14	6, 8, 13	mpbir2and 726	. 2 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
15	5, 14	pm2.61dan 825	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝑥 → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  {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  ax-nul 5259

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 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-sn 4585  df-iun 4952  df-disj 5070

This theorem is used by:  carsggect  34997