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Theorem unssdif 3317
 Description: Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
Assertion
Ref Expression
unssdif

Proof of Theorem unssdif
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2693 . . . . . . . 8
2 eldif 3086 . . . . . . . 8
31, 2mpbiran 925 . . . . . . 7
43anbi1i 454 . . . . . 6
5 eldif 3086 . . . . . 6
6 ioran 742 . . . . . 6
74, 5, 63bitr4i 211 . . . . 5
87biimpi 119 . . . 4
98con2i 617 . . 3
10 elun 3223 . . 3
11 eldif 3086 . . . 4
121, 11mpbiran 925 . . 3
139, 10, 123imtr4i 200 . 2
1413ssriv 3107 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 103   wo 698   wcel 1481  cvv 2690   cdif 3074   cun 3075   wss 3077 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-v 2692  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090 This theorem is referenced by: (None)
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