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Theorem disjun0 32615
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 4813 . . . . 5 (∅ ∈ 𝐴 → {∅} ⊆ 𝐴)
2 ssequn2 4199 . . . . 5 ({∅} ⊆ 𝐴 ↔ (𝐴 ∪ {∅}) = 𝐴)
31, 2sylib 218 . . . 4 (∅ ∈ 𝐴 → (𝐴 ∪ {∅}) = 𝐴)
43disjeq1d 5123 . . 3 (∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥Disj 𝑥𝐴 𝑥))
54biimparc 479 . 2 ((Disj 𝑥𝐴 𝑥 ∧ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
6 simpl 482 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥𝐴 𝑥)
7 in0 4401 . . . 4 ( 𝑥𝐴 𝑥 ∩ ∅) = ∅
87a1i 11 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)
9 0ex 5313 . . . . 5 ∅ ∈ V
10 id 22 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
1110disjunsn 32614 . . . . 5 ((∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
129, 11mpan 690 . . . 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 713 . 2 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
155, 14pm2.61dan 813 1 (Disj 𝑥𝐴 𝑥Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631   ciun 4996  Disj wdisj 5115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-iun 4998  df-disj 5116
This theorem is referenced by:  carsggect  34300
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