Theorem onunisuc 6396

Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6406. (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 6395	. 2 ⊢ (𝐴 ∈ On → Tr 𝐴)
2		unisucg 6365	. 2 ⊢ (𝐴 ∈ On → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
3	1, 2	mpbid 231	1 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ∪ cuni 4849  Tr wtr 5203  Oncon0 6288  suc csuc 6290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-v 3442  df-un 3901  df-in 3903  df-ss 3913  df-sn 4571  df-pr 4573  df-uni 4850  df-tr 5204  df-po 5520  df-so 5521  df-fr 5562  df-we 5564  df-ord 6291  df-on 6292  df-suc 6294

This theorem is referenced by:  onunisuci  6406  nlimsuc  41287