Theorem onunisuci 6365

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	. . 3 ⊢ 𝐴 ∈ On
2	1	ontrci 6357	. 2 ⊢ Tr 𝐴
3	1	elexi 3441	. . 3 ⊢ 𝐴 ∈ V
4	3	unisuc 6327	. 2 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)
5	2, 4	mpbi 229	1 ⊢ ∪ suc 𝐴 = 𝐴

Syntax hints:   = wceq 1539   ∈ wcel 2108  ∪ cuni 4836  Tr wtr 5187  Oncon0 6251  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-v 3424  df-un 3888  df-in 3890  df-ss 3900  df-sn 4559  df-pr 4561  df-uni 4837  df-tr 5188  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-suc 6257

This theorem is referenced by:  rankuni  9552  onsucconni  34553  onsucsuccmpi  34559  finxp1o  35490