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Theorem ssnel 4494
 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 4466 . 2 ¬ 𝐵𝐵
2 ssel 3097 . 2 (𝐴𝐵 → (𝐵𝐴𝐵𝐵))
31, 2mtoi 654 1 (𝐴𝐵 → ¬ 𝐵𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1481   ⊆ 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  ax-setind 4462 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-v 2692  df-dif 3079  df-in 3083  df-ss 3090  df-sn 3539 This theorem is referenced by:  nntri1  6403  pw1ne3  7108  3nelsucpw1  7112  3nsssucpw1  7114
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