Theorem onunisuc 6419

Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6428. (Revised by BJ, 28-Dec-2024.)

Assertion

Ref	Expression
onunisuc	⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)

Proof of Theorem onunisuc

Step	Hyp	Ref	Expression
1		ontr 6418	. 2 ⊢ (𝐴 ∈ On → Tr 𝐴)
2		unisucg 6387	. 2 ⊢ (𝐴 ∈ On → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
3	1, 2	mpbid 232	1 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∪ cuni 4858  Tr wtr 5199  Oncon0 6307  suc csuc 6309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-v 3438  df-un 3908  df-ss 3920  df-sn 4578  df-pr 4580  df-uni 4859  df-tr 5200  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-ord 6310  df-on 6311  df-suc 6313

This theorem is referenced by:  onunisuci  6428  nlimsuc  43414