Theorem onunisuci 6284

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 6276	. 2 ⊢ Tr 𝐴
3	1	elexi 3430	. . 3 ⊢ 𝐴 ∈ V
4	3	unisuc 6246	. 2 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)
5	2, 4	mpbi 233	1 ⊢ ∪ suc 𝐴 = 𝐴

Syntax hints:   = wceq 1539   ∈ wcel 2112  ∪ cuni 4799  Tr wtr 5139  Oncon0 6170  suc csuc 6172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-v 3412  df-un 3864  df-in 3866  df-ss 3876  df-sn 4524  df-pr 4526  df-uni 4800  df-tr 5140  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-ord 6173  df-on 6174  df-suc 6176

This theorem is referenced by:  rankuni  9318  onsucconni  34168  onsucsuccmpi  34174  finxp1o  35082