Theorem onunisuc 6478

Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6487. (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 6477	. 2 ⊢ (𝐴 ∈ On → Tr 𝐴)
2		unisucg 6446	. 2 ⊢ (𝐴 ∈ On → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
3	1, 2	mpbid 235	1 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∪ cuni 4866  Tr wtr 5211  Oncon0 6365  suc csuc 6367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-ss 3897  df-un 4150  df-sn 4585  df-pr 4587  df-uni 4867  df-tr 5212  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6368  df-on 6369  df-suc 6371

This theorem is used by:  onunisuci  6487  ordunisuc  7847  nlimsuc  44579