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Theorem disjxsn 3927
 Description: A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjxsn Disj
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem disjxsn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfdisj2 3908 . 2 Disj
2 moeq 2859 . . 3
3 elsni 3545 . . . . 5
43adantr 274 . . . 4
54moimi 2064 . . 3
62, 5ax-mp 5 . 2
71, 6mpgbir 1429 1 Disj
 Colors of variables: wff set class Syntax hints:   wa 103   wceq 1331   wcel 1480  wmo 2000  csn 3527  Disj wdisj 3906 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rmo 2424  df-v 2688  df-sn 3533  df-disj 3907 This theorem is referenced by:  disjx0  3928
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