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Theorem inssun 3221
 Description: Intersection in terms of class difference and union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
Assertion
Ref Expression
inssun

Proof of Theorem inssun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pm3.1 704 . . . . 5
2 eldifn 3106 . . . . . 6
3 eldifn 3106 . . . . . 6
42, 3orim12i 709 . . . . 5
51, 4nsyl 591 . . . 4
6 elun 3124 . . . 4
75, 6sylnibr 635 . . 3
8 elin 3166 . . 3
9 vex 2613 . . . 4
10 eldif 2992 . . . 4
119, 10mpbiran 882 . . 3
127, 8, 113imtr4i 199 . 2
1312ssriv 3013 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 102   wo 662   wcel 1434  cvv 2610   cdif 2980   cun 2981   cin 2982   wss 2983 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996 This theorem is referenced by: (None)
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