Theorem disjun0 33171

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 4746	. . . . 5 ⊢ (∅ ∈ 𝐴 → {∅} ⊆ 𝐴)
2		ssequn2 4135	. . . . 5 ⊢ ({∅} ⊆ 𝐴 ↔ (𝐴 ∪ {∅}) = 𝐴)
3	1, 2	sylib 221	. . . 4 ⊢ (∅ ∈ 𝐴 → (𝐴 ∪ {∅}) = 𝐴)
4	3	disjeq1d 5078	. . 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 4345	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅
8	7	a1i 11	. . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅)
9		0ex 5261	. . . . 5 ⊢ ∅ ∈ V
10		id 23	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
11	10	disjunsn 33170	. . . . 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 3451   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  {csn 4584  ∪ ciun 4951  Disj wdisj 5070

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 2213  ax-ext 2733  ax-nul 5260

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 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rmo 3366  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-sn 4585  df-iun 4953  df-disj 5071

This theorem is used by:  carsggect  34933