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Theorem inssdif0im 3457
 Description: Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
Assertion
Ref Expression
inssdif0im

Proof of Theorem inssdif0im
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3286 . . . . . 6
21imbi1i 237 . . . . 5
3 imanim 678 . . . . 5
42, 3sylbi 120 . . . 4
5 eldif 3107 . . . . . 6
65anbi2i 453 . . . . 5
7 elin 3286 . . . . 5
8 anass 399 . . . . 5
96, 7, 83bitr4ri 212 . . . 4
104, 9sylnib 666 . . 3
1110alimi 1432 . 2
12 dfss2 3113 . 2
13 eq0 3408 . 2
1411, 12, 133imtr4i 200 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103  wal 1330   wceq 1332   wcel 2125   cdif 3095   cin 3097   wss 3098  c0 3390 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-dif 3100  df-in 3104  df-ss 3111  df-nul 3391 This theorem is referenced by:  disjdif  3462
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