Theorem disjun0 33076

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 4749	. . . . 5 ⊢ (∅ ∈ 𝐴 → {∅} ⊆ 𝐴)
2		ssequn2 4138	. . . . 5 ⊢ ({∅} ⊆ 𝐴 ↔ (𝐴 ∪ {∅}) = 𝐴)
3	1, 2	sylib 221	. . . 4 ⊢ (∅ ∈ 𝐴 → (𝐴 ∪ {∅}) = 𝐴)
4	3	disjeq1d 5082	. . 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 4348	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅
8	7	a1i 11	. . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅)
9		0ex 5268	. . . . 5 ⊢ ∅ ∈ V
10		id 23	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
11	10	disjunsn 33075	. . . . 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 1570   ∈ wcel 2145  Vcvv 3453   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  {csn 4587  ∪ ciun 4954  Disj wdisj 5074

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rmo 3367  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-sn 4588  df-iun 4956  df-disj 5075

This theorem is used by:  carsggect  34837