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Theorem disjx0 4679
 Description: An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjx0 Disj 𝑥 ∈ ∅ 𝐵

Proof of Theorem disjx0
StepHypRef Expression
1 0ss 4005 . 2 ∅ ⊆ {∅}
2 disjxsn 4678 . 2 Disj 𝑥 ∈ {∅}𝐵
3 disjss1 4658 . 2 (∅ ⊆ {∅} → (Disj 𝑥 ∈ {∅}𝐵Disj 𝑥 ∈ ∅ 𝐵))
41, 2, 3mp2 9 1 Disj 𝑥 ∈ ∅ 𝐵
 Colors of variables: wff setvar class Syntax hints:   ⊆ wss 3607  ∅c0 3948  {csn 4210  Disj wdisj 4652 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rmo 2949  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-disj 4653 This theorem is referenced by: (None)
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