Theorem sssucid 4512

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 3370	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝐴})
2		df-suc 4468	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
3	1, 2	sseqtrri 3262	1 ⊢ 𝐴 ⊆ suc 𝐴

Syntax hints:   ∪ cun 3198   ⊆ wss 3200  {csn 3669  suc csuc 4462

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-suc 4468

This theorem is referenced by:  trsuc  4519  ordsuc  4661  0elnn  4717  sucinc  6612  sucinc2  6613  oasuc  6631  phplem4  7040  phplem4dom  7047  phplem4on  7053  fiintim  7122  fidcenumlemrk  7152  fidcenumlemr  7153  bj-nntrans  16546