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Theorem 0disj 3790
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 3289 . . 3 ∅ ⊆ {𝑥}
21rgenw 2419 . 2 𝑥𝐴 ∅ ⊆ {𝑥}
3 sndisj 3789 . 2 Disj 𝑥𝐴 {𝑥}
4 disjss2 3777 . 2 (∀𝑥𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥𝐴 {𝑥} → Disj 𝑥𝐴 ∅))
52, 3, 4mp2 16 1 Disj 𝑥𝐴
Colors of variables: wff set class
Syntax hints:  wral 2349  wss 2974  c0 3258  {csn 3406  Disj wdisj 3774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rmo 2357  df-v 2604  df-dif 2976  df-in 2980  df-ss 2987  df-nul 3259  df-sn 3412  df-disj 3775
This theorem is referenced by: (None)
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