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Theorem sssucid 4393
Description: A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)
Assertion
Ref Expression
sssucid 𝐴 ⊆ suc 𝐴

Proof of Theorem sssucid
StepHypRef Expression
1 ssun1 3285 . 2 𝐴 ⊆ (𝐴 ∪ {𝐴})
2 df-suc 4349 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
31, 2sseqtrri 3177 1 𝐴 ⊆ suc 𝐴
Colors of variables: wff set class
Syntax hints:  cun 3114  wss 3116  {csn 3576  suc csuc 4343
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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-suc 4349
This theorem is referenced by:  trsuc  4400  ordsuc  4540  0elnn  4596  sucinc  6413  sucinc2  6414  oasuc  6432  phplem4  6821  phplem4dom  6828  phplem4on  6833  fiintim  6894  fidcenumlemrk  6919  fidcenumlemr  6920  bj-nntrans  13833
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