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Theorem 0disj 5024
 Description: Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
0disj Disj 𝑥𝐴

Proof of Theorem 0disj
StepHypRef Expression
1 0ss 4292 . . 3 ∅ ⊆ {𝑥}
21rgenw 3082 . 2 𝑥𝐴 ∅ ⊆ {𝑥}
3 sndisj 5023 . 2 Disj 𝑥𝐴 {𝑥}
4 disjss2 5000 . 2 (∀𝑥𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥𝐴 {𝑥} → Disj 𝑥𝐴 ∅))
52, 3, 4mp2 9 1 Disj 𝑥𝐴
 Colors of variables: wff setvar class Syntax hints:  ∀wral 3070   ⊆ wss 3858  ∅c0 4225  {csn 4522  Disj wdisj 4997 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-mo 2557  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rmo 3078  df-v 3411  df-dif 3861  df-in 3865  df-ss 3875  df-nul 4226  df-sn 4523  df-disj 4998 This theorem is referenced by: (None)
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