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Theorem reldisj 3594
 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 3262 . . . 4
2 pm5.44 877 . . . . . 6
3 eldif 3221 . . . . . . 7
43imbi2i 303 . . . . . 6
52, 4syl6bbr 254 . . . . 5
65sps 1754 . . . 4
71, 6sylbi 187 . . 3
87albidv 1625 . 2
9 disj1 3593 . 2
10 dfss2 3262 . 2
118, 9, 103bitr4g 279 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wa 358  wal 1540   wceq 1642   wcel 1710   cdif 3206   cin 3208   wss 3257  c0 3550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by:  disj2  3598
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