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Theorem 0disj 4575
 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 3924 . . 3 ∅ ⊆ {𝑥}
21rgenw 2908 . 2 𝑥𝐴 ∅ ⊆ {𝑥}
3 sndisj 4574 . 2 Disj 𝑥𝐴 {𝑥}
4 disjss2 4556 . 2 (∀𝑥𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥𝐴 {𝑥} → Disj 𝑥𝐴 ∅))
52, 3, 4mp2 9 1 Disj 𝑥𝐴
 Colors of variables: wff setvar class Syntax hints:  ∀wral 2896   ⊆ wss 3540  ∅c0 3874  {csn 4125  Disj wdisj 4553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rmo 2904  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-disj 4554 This theorem is referenced by: (None)
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