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Theorem disjun0 29250
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 4308 . . . . 5 (∅ ∈ 𝐴 → {∅} ⊆ 𝐴)
2 ssequn2 3764 . . . . 5 ({∅} ⊆ 𝐴 ↔ (𝐴 ∪ {∅}) = 𝐴)
31, 2sylib 208 . . . 4 (∅ ∈ 𝐴 → (𝐴 ∪ {∅}) = 𝐴)
43disjeq1d 4591 . . 3 (∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥Disj 𝑥𝐴 𝑥))
54biimparc 504 . 2 ((Disj 𝑥𝐴 𝑥 ∧ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
6 simpl 473 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥𝐴 𝑥)
7 in0 3940 . . . 4 ( 𝑥𝐴 𝑥 ∩ ∅) = ∅
87a1i 11 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)
9 0ex 4750 . . . . 5 ∅ ∈ V
10 id 22 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
1110disjunsn 29249 . . . . 5 ((∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
129, 11mpan 705 . . . 4 (¬ ∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
1312adantl 482 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
146, 8, 13mpbir2and 956 . 2 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
155, 14pm2.61dan 831 1 (Disj 𝑥𝐴 𝑥Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cun 3553  cin 3554  wss 3555  c0 3891  {csn 4148   ciun 4485  Disj wdisj 4583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-iun 4487  df-disj 4584
This theorem is referenced by:  carsggect  30158
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