Theorem onunisuc 6389

Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6399. (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 6388	. 2 ⊢ (𝐴 ∈ On → Tr 𝐴)
2		unisucg 6358	. 2 ⊢ (𝐴 ∈ On → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
3	1, 2	mpbid 231	1 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  ∪ cuni 4844  Tr wtr 5198  Oncon0 6281  suc csuc 6283

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-v 3439  df-un 3897  df-in 3899  df-ss 3909  df-sn 4566  df-pr 4568  df-uni 4845  df-tr 5199  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-ord 6284  df-on 6285  df-suc 6287

This theorem is referenced by:  onunisuci  6399  nlimsuc  41270