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Theorem ssnel 4227
Description: Relationship between subset and elementhood. In the context of ordinals this can be seen as an ordering law. (Contributed by Jim Kingdon, 22-Jul-2019.)
Assertion
Ref Expression
ssnel (AB → ¬ B A)

Proof of Theorem ssnel
StepHypRef Expression
1 elirr 4206 . 2 ¬ B B
2 ssel 2915 . 2 (AB → (B AB B))
31, 2mtoi 577 1 (AB → ¬ B A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wcel 1375  wss 2893
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-setind 4202
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ne 2189  df-ral 2288  df-v 2536  df-dif 2896  df-in 2900  df-ss 2907  df-sn 3355
This theorem is referenced by:  nntri1  5984
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