Theorem sssucid 4152

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 3106	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝐴})
2		df-suc 4108	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
3	1, 2	sseqtr4i 2978	1 ⊢ 𝐴 ⊆ suc 𝐴

Syntax hints:   ∪ cun 2915   ⊆ wss 2917  {csn 3375  suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-suc 4108

This theorem is referenced by:  trsuc  4159  ordsuc  4287  onpsssuc  4295  0elnn  4340  sucinc  6025  sucinc2  6026  oasuc  6044  phplem4  6318  phplem4dom  6324  phplem4on  6329  bj-nntrans  10076