Theorem sssucid 3037

Description: A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes).

Assertion

Ref	Expression
sssucid	|- A (_ suc A

Proof of Theorem sssucid

Step	Hyp	Ref	Expression
1		ssun1 2183	. 2 |- A (_ (A u. {A})
2		df-suc 2944	. 2 |- suc A = (A u. {A})
3	1, 2	sseqtr4 2084	1 |- A (_ suc A

Syntax hints:   u. cun 2035   (_ wss 2037  {csn 2399  suc csuc 2940

This theorem is referenced by:  suceloni 3052  limsssuc 3111  oaordi 4164  oelim2 4206  phplem4 4491  php 4493  onomeneq 4498  unifi 4532  fiint 4534  fodomfi 4540  r1pwcl 4659  ranksuc 4672

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-in 2041  df-ss 2043  df-suc 2944

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