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Theorem ssnel 4263
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 → ¬ 𝐵𝐴)

Proof of Theorem ssnel
StepHypRef Expression
1 elirr 4236 . 2 ¬ 𝐵𝐵
2 ssel 2936 . 2 (𝐴𝐵 → (𝐵𝐴𝐵𝐵))
31, 2mtoi 590 1 (𝐴𝐵 → ¬ 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1393  wss 2914
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  ax-setind 4232
This theorem depends on definitions:  df-bi 110  df-3an 887  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-ral 2308  df-v 2556  df-dif 2917  df-in 2921  df-ss 2928  df-sn 3378
This theorem is referenced by:  nntri1  6037
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