Theorem sssucid 4518

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 3372	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝐴})
2		df-suc 4474	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
3	1, 2	sseqtrri 3263	1 ⊢ 𝐴 ⊆ suc 𝐴

Syntax hints:   ∪ cun 3199   ⊆ wss 3201  {csn 3673  suc csuc 4468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-suc 4474

This theorem is referenced by:  trsuc  4525  ordsuc  4667  0elnn  4723  sucinc  6656  sucinc2  6657  oasuc  6675  phplem4  7084  phplem4dom  7091  phplem4on  7097  fiintim  7166  fidcenumlemrk  7196  fidcenumlemr  7197  bj-nntrans  16650