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Theorem reldisj 3420
 Description: Two ways of saying that two classes are disjoint, using the complement of relative to a universe . (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
reldisj

Proof of Theorem reldisj
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfss2 3092 . . . 4
2 pm5.44 911 . . . . . 6
3 eldif 3086 . . . . . . 7
43imbi2i 225 . . . . . 6
52, 4bitr4di 197 . . . . 5
65sps 1518 . . . 4
71, 6sylbi 120 . . 3
87albidv 1797 . 2
9 disj1 3419 . 2
10 dfss2 3092 . 2
118, 9, 103bitr4g 222 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wb 104  wal 1330   wceq 1332   wcel 1481   cdif 3074   cin 3076   wss 3077  c0 3369 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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2692  df-dif 3079  df-in 3083  df-ss 3090  df-nul 3370 This theorem is referenced by:  disj2  3424  ssdifsn  3660  structcnvcnv  12034
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