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Theorem 0disj 5084
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 4343 . . 3 ∅ ⊆ {𝑥}
21rgenw 3065 . 2 𝑥𝐴 ∅ ⊆ {𝑥}
3 sndisj 5083 . 2 Disj 𝑥𝐴 {𝑥}
4 disjss2 5060 . 2 (∀𝑥𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥𝐴 {𝑥} → Disj 𝑥𝐴 ∅))
52, 3, 4mp2 9 1 Disj 𝑥𝐴
Colors of variables: wff setvar class
Syntax hints:  wral 3061  wss 3898  c0 4269  {csn 4573  Disj wdisj 5057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rmo 3349  df-v 3443  df-dif 3901  df-in 3905  df-ss 3915  df-nul 4270  df-sn 4574  df-disj 5058
This theorem is referenced by: (None)
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