Theorem onunisuci 6179

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 6171	. 2 ⊢ Tr 𝐴
3	1	elexi 3456	. . 3 ⊢ 𝐴 ∈ V
4	3	unisuc 6142	. 2 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)
5	2, 4	mpbi 231	1 ⊢ ∪ suc 𝐴 = 𝐴

Syntax hints:   = wceq 1522   ∈ wcel 2081  ∪ cuni 4745  Tr wtr 5063  Oncon0 6066  suc csuc 6068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-v 3439  df-un 3864  df-in 3866  df-ss 3874  df-sn 4473  df-pr 4475  df-uni 4746  df-tr 5064  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-ord 6069  df-on 6070  df-suc 6072

This theorem is referenced by:  rankuni  9138  onsucconni  33394  onsucsuccmpi  33400  finxp1o  34204