Theorem onunisuc 6493

Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6503. (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 6492	. 2 ⊢ (𝐴 ∈ On → Tr 𝐴)
2		unisucg 6461	. 2 ⊢ (𝐴 ∈ On → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
3	1, 2	mpbid 232	1 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ∪ cuni 4906  Tr wtr 5258  Oncon0 6383  suc csuc 6385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-v 3481  df-un 3955  df-ss 3967  df-sn 4626  df-pr 4628  df-uni 4907  df-tr 5259  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-suc 6389

This theorem is referenced by:  onunisuci  6503  nlimsuc  43459