Theorem onunisuci 6290

Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)

Hypothesis

Ref	Expression
on.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
onunisuci	⊢ ∪ suc 𝐴 = 𝐴

Proof of Theorem onunisuci

Step	Hyp	Ref	Expression
1		on.1	. . 3 ⊢ 𝐴 ∈ On
2	1	ontrci 6282	. 2 ⊢ Tr 𝐴
3	1	elexi 3505	. . 3 ⊢ 𝐴 ∈ V
4	3	unisuc 6253	. 2 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)
5	2, 4	mpbi 232	1 ⊢ ∪ suc 𝐴 = 𝐴

Syntax hints:   = wceq 1537   ∈ wcel 2114  ∪ cuni 4824  Tr wtr 5158  Oncon0 6177  suc csuc 6179

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3488  df-un 3929  df-in 3931  df-ss 3940  df-sn 4554  df-pr 4556  df-uni 4825  df-tr 5159  df-po 5460  df-so 5461  df-fr 5500  df-we 5502  df-ord 6180  df-on 6181  df-suc 6183

This theorem is referenced by:  rankuni  9278  onsucconni  33792  onsucsuccmpi  33798  finxp1o  34689