Theorem onunisuci 6483

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 6474	. 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 2145  ∪ cuni 4870  Oncon0 6361  suc csuc 6363

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 2147  ax-9 2155  ax-ext 2734

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-v 3455  df-un 3907  df-ss 3919  df-sn 4588  df-pr 4590  df-uni 4871  df-tr 5217  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6364  df-on 6365  df-suc 6367

This theorem is used by:  rankuni  9848  onsucconni  37058  onsucsuccmpi  37064  finxp1o  38148