Theorem sssucid 3742

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 C_ suc A

Proof of Theorem sssucid

Step	Hyp	Ref	Expression
1		ssun1 2767	. 2 |- A C_ (A u. {A})
2		df-suc 3663	. 2 |- suc A = (A u. {A})
3	1, 2	sseqtr4i 2650	1 |- A C_ suc A

Syntax hints:   u. cun 2591   C_ wss 2593  {csn 3044  suc csuc 3659

This theorem is referenced by:  trsuc 3752  suceloni 3894  limsssuc 3934  oaordi 5227  oelim2 5270  ac6sfilem3 5508  ac6sfi 5509  phplem4 5605  php 5607  onomeneq 5612  unifi 5648  fiint 5650  fodomfi 5656  r1pwcl 5798  ranksuc 5811  fbssint 10279  axfelem10 14040  axfelem15 14045  top2usne 14898  finsschain 15373  fcluscomplem 15620  suctrALT2VD 16660  suctrALT2 16661

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-in 2603  df-ss 2605  df-suc 3663

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