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Theorem 0disj 4002
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 3463 . . 3 ∅ ⊆ {𝑥}
21rgenw 2532 . 2 𝑥𝐴 ∅ ⊆ {𝑥}
3 sndisj 4001 . 2 Disj 𝑥𝐴 {𝑥}
4 disjss2 3985 . 2 (∀𝑥𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥𝐴 {𝑥} → Disj 𝑥𝐴 ∅))
52, 3, 4mp2 16 1 Disj 𝑥𝐴
Colors of variables: wff set class
Syntax hints:  wral 2455  wss 3131  c0 3424  {csn 3594  Disj wdisj 3982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rmo 2463  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-disj 3983
This theorem is referenced by: (None)
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