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Theorem unssdif 3172
 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 2560 . . . . . . . 8
2 eldif 2927 . . . . . . . 8
31, 2mpbiran 847 . . . . . . 7
43anbi1i 431 . . . . . 6
5 eldif 2927 . . . . . 6
6 ioran 669 . . . . . 6
74, 5, 63bitr4i 201 . . . . 5
87biimpi 113 . . . 4
98con2i 557 . . 3
10 elun 3084 . . 3
11 eldif 2927 . . . 4
121, 11mpbiran 847 . . 3
139, 10, 123imtr4i 190 . 2
1413ssriv 2949 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 97   wo 629   wcel 1393  cvv 2557   cdif 2914   cun 2915   wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
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