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Theorem disjun0 30339
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 4734 . . . . 5 (∅ ∈ 𝐴 → {∅} ⊆ 𝐴)
2 ssequn2 4158 . . . . 5 ({∅} ⊆ 𝐴 ↔ (𝐴 ∪ {∅}) = 𝐴)
31, 2sylib 220 . . . 4 (∅ ∈ 𝐴 → (𝐴 ∪ {∅}) = 𝐴)
43disjeq1d 5031 . . 3 (∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥Disj 𝑥𝐴 𝑥))
54biimparc 482 . 2 ((Disj 𝑥𝐴 𝑥 ∧ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
6 simpl 485 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥𝐴 𝑥)
7 in0 4344 . . . 4 ( 𝑥𝐴 𝑥 ∩ ∅) = ∅
87a1i 11 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)
9 0ex 5203 . . . . 5 ∅ ∈ V
10 id 22 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
1110disjunsn 30338 . . . . 5 ((∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
129, 11mpan 688 . . . 4 (¬ ∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
1312adantl 484 . . 3 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥𝐴 𝑥 ∧ ( 𝑥𝐴 𝑥 ∩ ∅) = ∅)))
146, 8, 13mpbir2and 711 . 2 ((Disj 𝑥𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
155, 14pm2.61dan 811 1 (Disj 𝑥𝐴 𝑥Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  Vcvv 3494  cun 3933  cin 3934  wss 3935  c0 4290  {csn 4560   ciun 4911  Disj wdisj 5023
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-sn 4561  df-iun 4913  df-disj 5024
This theorem is referenced by:  carsggect  31571
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