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Theorem disjxsn 4206
 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 4184 . 2 Disj
2 moeq 3110 . . 3
3 elsni 3838 . . . . 5
43adantr 452 . . . 4
54moimi 2328 . . 3
62, 5ax-mp 8 . 2
71, 6mpgbir 1559 1 Disj
 Colors of variables: wff set class Syntax hints:   wa 359   wceq 1652   wcel 1725  wmo 2282  csn 3814  Disj wdisj 4182 This theorem is referenced by:  disjx0  4207  disjdifprg  24017 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rmo 2713  df-v 2958  df-sn 3820  df-disj 4183
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