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Theorem disjx0 5138
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 4394 . 2 ∅ ⊆ {∅}
2 disjxsn 5137 . 2 Disj 𝑥 ∈ {∅}𝐵
3 disjss1 5115 . 2 (∅ ⊆ {∅} → (Disj 𝑥 ∈ {∅}𝐵Disj 𝑥 ∈ ∅ 𝐵))
41, 2, 3mp2 9 1 Disj 𝑥 ∈ ∅ 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3946  c0 4320  {csn 4624  Disj wdisj 5109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-rmo 3377  df-v 3477  df-dif 3949  df-in 3953  df-ss 3963  df-nul 4321  df-sn 4625  df-disj 5110
This theorem is referenced by: (None)
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