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Theorem unssin 3310
 Description: Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
Assertion
Ref Expression
unssin

Proof of Theorem unssin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 oranim 770 . . . . 5
2 eldifn 3194 . . . . . 6
3 eldifn 3194 . . . . . 6
42, 3anim12i 336 . . . . 5
51, 4nsyl 617 . . . 4
6 elin 3254 . . . 4
75, 6sylnibr 666 . . 3
8 elun 3212 . . 3
9 vex 2684 . . . 4
10 eldif 3075 . . . 4
119, 10mpbiran 924 . . 3
127, 8, 113imtr4i 200 . 2
1312ssriv 3096 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 103   wo 697   wcel 1480  cvv 2681   cdif 3063   cun 3064   cin 3065   wss 3066 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 603  ax-in2 604  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 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079 This theorem is referenced by: (None)
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