Theorem onunisuc 6422

Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6431. (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 6421	. 2 ⊢ (𝐴 ∈ On → Tr 𝐴)
2		unisucg 6390	. 2 ⊢ (𝐴 ∈ On → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
3	1, 2	mpbid 233	1 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ∪ cuni 4838  Tr wtr 5179  Oncon0 6310  suc csuc 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-v 3433  df-un 3888  df-ss 3900  df-sn 4556  df-pr 4558  df-uni 4839  df-tr 5180  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-ord 6313  df-on 6314  df-suc 6316

This theorem is referenced by:  onunisuci  6431  nlimsuc  43885