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Theorem disjun0 29925
Description: Adding the empty element preserves disjointness. (Contributed by Thierry Arnoux, 30-May-2020.)
Assertion
Ref Expression
disjun0 (Disj 𝑥𝐴 𝑥Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem disjun0
StepHypRef Expression
1 snssi 4527 . . . . 5 (∅ ∈ 𝐴 → {∅} ⊆ 𝐴)
2 ssequn2 3984 . . . . 5 ({∅} ⊆ 𝐴 ↔ (𝐴 ∪ {∅}) = 𝐴)
31, 2sylib 210 . . . 4 (∅ ∈ 𝐴 → (𝐴 ∪ {∅}) = 𝐴)
43disjeq1d 4819 . . 3 (∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥Disj 𝑥𝐴 𝑥))
54biimparc 472 . 2 ((Disj 𝑥𝐴 𝑥 ∧ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
6 simpl 475 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥𝐴 𝑥)
7 in0 4164 . . . 4 ( 𝑥𝐴 𝑥 ∩ ∅) = ∅
87a1i 11 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)
9 0ex 4984 . . . . 5 ∅ ∈ V
10 id 22 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
1110disjunsn 29924 . . . . 5 ((∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
129, 11mpan 682 . . . 4 (¬ ∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
1312adantl 474 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
146, 8, 13mpbir2and 705 . 2 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
155, 14pm2.61dan 848 1 (Disj 𝑥𝐴 𝑥Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  cun 3767  cin 3768  wss 3769  c0 4115  {csn 4368   ciun 4710  Disj wdisj 4811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-sn 4369  df-iun 4712  df-disj 4812
This theorem is referenced by:  carsggect  30896
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