Theorem onunisuci 6298

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 6290	. 2 ⊢ Tr 𝐴
3	1	elexi 3513	. . 3 ⊢ 𝐴 ∈ V
4	3	unisuc 6261	. 2 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)
5	2, 4	mpbi 232	1 ⊢ ∪ suc 𝐴 = 𝐴

Syntax hints:   = wceq 1533   ∈ wcel 2110  ∪ cuni 4831  Tr wtr 5164  Oncon0 6185  suc csuc 6187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-un 3940  df-in 3942  df-ss 3951  df-sn 4561  df-pr 4563  df-uni 4832  df-tr 5165  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-ord 6188  df-on 6189  df-suc 6191

This theorem is referenced by:  rankuni  9286  onsucconni  33780  onsucsuccmpi  33786  finxp1o  34667