Theorem onunisuci 6469

Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)

Hypothesis

Ref	Expression
on.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
onunisuci	⊢ ∪ suc 𝐴 = 𝐴

Proof of Theorem onunisuci

Step	Hyp	Ref	Expression
1		on.1	. 2 ⊢ 𝐴 ∈ On
2		onunisuc 6460	. 2 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∪ suc 𝐴 = 𝐴

Syntax hints:   = wceq 1562   ∈ wcel 2144  ∪ cuni 4867  Oncon0 6348  suc csuc 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-v 3458  df-un 3911  df-ss 3923  df-sn 4585  df-pr 4587  df-uni 4868  df-tr 5210  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-ord 6351  df-on 6352  df-suc 6354

This theorem is referenced by:  rankuni  9823  onsucconni  36802  onsucsuccmpi  36808  finxp1o  37891