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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 4397 . 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 3947  c0 4323  {csn 4629  Disj wdisj 5113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2530  df-clab 2706  df-cleq 2720  df-clel 2806  df-rmo 3373  df-v 3473  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4324  df-sn 4630  df-disj 5114
This theorem is referenced by: (None)
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