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Theorem sssucid 4179
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 3133 . 2 𝐴 ⊆ (𝐴 ∪ {𝐴})
2 df-suc 4135 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
31, 2sseqtr4i 3005 1 𝐴 ⊆ suc 𝐴
Colors of variables: wff set class
Syntax hints:  cun 2942  wss 2944  {csn 3402  suc csuc 4129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-suc 4135
This theorem is referenced by:  trsuc  4186  ordsuc  4314  onpsssuc  4322  0elnn  4367  sucinc  6055  sucinc2  6056  oasuc  6074  phplem4  6348  phplem4dom  6354  phplem4on  6359  bj-nntrans  10442
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