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Theorem difsnpssim 3507
 Description: is a proper subclass of if is a member of . In classical logic, the converse holds as well. (Contributed by Jim Kingdon, 9-Aug-2018.)
Assertion
Ref Expression
difsnpssim

Proof of Theorem difsnpssim
StepHypRef Expression
1 notnot 559 . 2
2 difss 3070 . . . 4
32biantrur 287 . . 3
4 difsnb 3506 . . . 4
54necon3bbii 2242 . . 3
6 df-pss 2933 . . 3
73, 5, 63bitr4i 201 . 2
81, 7sylib 127 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 97   wcel 1393   wne 2204   cdif 2914   wss 2917   wpss 2918  csn 3375 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-ne 2206  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-pss 2933  df-sn 3381 This theorem is referenced by: (None)
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