Theorem sssucid 3047

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 2193	. 2 |- A (_ (A u. {A})
2		df-suc 2954	. 2 |- suc A = (A u. {A})
3	1, 2	sseqtr4 2094	1 |- A (_ suc A

Syntax hints:   u. cun 2045   (_ wss 2047  {csn 2409  suc csuc 2950

This theorem is referenced by:  suceloni 3062  limsssuc 3121  oaordi 4180  oelim2 4222  phplem4 4511  php 4513  onomeneq 4519  unifiOLD 4557  fiint 4559  fiintOLD 4560  fodomfiOLD 4566  r1pwcl 4687  ranksuc 4700  top2usne 10549

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-in 2051  df-ss 2053  df-suc 2954

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