Theorem onunisuci 6486

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 6477	. 2 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∪ suc 𝐴 = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  ∪ cuni 4875  Oncon0 6364  suc csuc 6366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-v 3460  df-un 3913  df-ss 3925  df-sn 4593  df-pr 4595  df-uni 4876  df-tr 5222  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-ord 6367  df-on 6368  df-suc 6370

This theorem is used by:  rankuni  9837  onsucconni  36980  onsucsuccmpi  36986  finxp1o  38070