Theorem sssucid 2300

Description: A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes).

Assertion

Ref	Expression
sssucid	⊢ A ⊆ suc A

Proof of Theorem sssucid

Step	Hyp	Ref	Expression
1		ssun1 1621	. 2 ⊢ A ⊆ (A ∪ {A})
2		df-suc 2205	. 2 ⊢ suc A = (A ∪ {A})
3	1, 2	sseqtr4 1533	1 ⊢ A ⊆ suc A

Syntax hints:   ∪ cun 1485   ⊆ wss 1487  {csn 1808  suc csuc 2201

This theorem is referenced by:  suceloni 2314  limsssuc 2362  oaordi 3148  phplem5 3407  php 3409  onomeneq 3414  fiint 3445  r1pwcl 3530

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-suc 2205

metamath.org