Theorem sssucid 5802

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 3776	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝐴})
2		df-suc 5729	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
3	1, 2	sseqtr4i 3638	1 ⊢ 𝐴 ⊆ suc 𝐴

Syntax hints:   ∪ cun 3572   ⊆ wss 3574  {csn 4177  suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-suc 5729

This theorem is referenced by:  trsuc  5810  suceloni  7013  limsssuc  7050  oaordi  7626  omeulem1  7662  oelim2  7675  nnaordi  7698  phplem4  8142  php  8144  onomeneq  8150  fiint  8237  cantnfval2  8566  cantnfle  8568  cantnfp1lem3  8577  cnfcomlem  8596  ranksuc  8728  fseqenlem1  8847  pwsdompw  9026  fin1a2lem12  9233  canthp1lem2  9475  nosupbday  31851  nosupbnd1  31860  nosupbnd2lem1  31861  limsucncmpi  32444  finxpreclem3  33230  clsk1independent  38344  suctrALT  39061  suctrALT2VD  39071  suctrALT2  39072  suctrALTcf  39158  suctrALTcfVD  39159  suctrALT3  39160