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Theorem disjun0 28790
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 4280 . . . . 5 (∅ ∈ 𝐴 → {∅} ⊆ 𝐴)
2 ssequn2 3748 . . . . 5 ({∅} ⊆ 𝐴 ↔ (𝐴 ∪ {∅}) = 𝐴)
31, 2sylib 207 . . . 4 (∅ ∈ 𝐴 → (𝐴 ∪ {∅}) = 𝐴)
43disjeq1d 4561 . . 3 (∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥Disj 𝑥𝐴 𝑥))
54biimparc 503 . 2 ((Disj 𝑥𝐴 𝑥 ∧ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
6 simpl 472 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥𝐴 𝑥)
7 in0 3920 . . . 4 ( 𝑥𝐴 𝑥 ∩ ∅) = ∅
87a1i 11 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)
9 0ex 4718 . . . . 5 ∅ ∈ V
10 id 22 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
1110disjunsn 28789 . . . . 5 ((∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
129, 11mpan 702 . . . 4 (¬ ∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
1312adantl 481 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
146, 8, 13mpbir2and 959 . 2 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
155, 14pm2.61dan 828 1 (Disj 𝑥𝐴 𝑥Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cun 3538  cin 3539  wss 3540  c0 3874  {csn 4125   ciun 4455  Disj wdisj 4553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-iun 4457  df-disj 4554
This theorem is referenced by:  carsggect  29707
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