Theorem sssucid 6434

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

Step	Hyp	Ref	Expression
1		ssun1 4153	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝐴})
2		df-suc 6358	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
3	1, 2	sseqtrri 4008	1 ⊢ 𝐴 ⊆ suc 𝐴

Syntax hints:   ∪ cun 3924   ⊆ wss 3926  {csn 4601  suc csuc 6354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-ss 3943  df-suc 6358

This theorem is referenced by:  trsuc  6441  sucexeloniOLD  7804  suceloniOLD  7806  limsssuc  7845  oaordi  8558  omeulem1  8594  oelim2  8607  nnaordi  8630  naddcllem  8688  phplem2  9219  php  9221  phpOLD  9231  onomeneqOLD  9238  enp1i  9285  fiint  9338  fiintOLD  9339  cantnfval2  9683  cantnfle  9685  cantnfp1lem3  9694  cnfcomlem  9713  ttrclss  9734  ranksuc  9879  fseqenlem1  10038  pwsdompw  10217  fin1a2lem12  10425  canthp1lem2  10667  nosupbnd1  27678  nosupbnd2lem1  27679  noinfbnd1  27693  noinfbnd2lem1  27694  satfvsucsuc  35387  satffunlem2lem2  35428  satffunlem2  35430  limsucncmpi  36463  finxpreclem3  37411  insucid  43427  minregex  43558  clsk1independent  44070  grur1cld  44256  suctrALT  44850  suctrALT2VD  44860  suctrALT2  44861  suctrALTcf  44946  suctrALTcfVD  44947  suctrALT3  44948