Theorem onunisuc 6470

Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6479. (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 6469	. 2 ⊢ (𝐴 ∈ On → Tr 𝐴)
2		unisucg 6438	. 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 1570   ∈ wcel 2145  ∪ cuni 4867  Tr wtr 5212  Oncon0 6357  suc csuc 6359

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-un 3904  df-ss 3916  df-sn 4585  df-pr 4587  df-uni 4868  df-tr 5213  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361  df-suc 6363

This theorem is used by:  onunisuci  6479  nlimsuc  44282