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Theorem disjx0 5142
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 4396 . 2 ∅ ⊆ {∅}
2 disjxsn 5141 . 2 Disj 𝑥 ∈ {∅}𝐵
3 disjss1 5119 . 2 (∅ ⊆ {∅} → (Disj 𝑥 ∈ {∅}𝐵Disj 𝑥 ∈ ∅ 𝐵))
41, 2, 3mp2 9 1 Disj 𝑥 ∈ ∅ 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3948  c0 4322  {csn 4628  Disj wdisj 5113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-rmo 3376  df-v 3476  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-disj 5114
This theorem is referenced by: (None)
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