Theorem sssucid 4332

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 3234	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝐴})
2		df-suc 4288	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
3	1, 2	sseqtrri 3127	1 ⊢ 𝐴 ⊆ suc 𝐴

Syntax hints:   ∪ cun 3064   ⊆ wss 3066  {csn 3522  suc csuc 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-suc 4288

This theorem is referenced by:  trsuc  4339  ordsuc  4473  0elnn  4527  sucinc  6334  sucinc2  6335  oasuc  6353  phplem4  6742  phplem4dom  6749  phplem4on  6754  fiintim  6810  fidcenumlemrk  6835  fidcenumlemr  6836  bj-nntrans  13138