Theorem onunisuci 6502

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	. 2 ⊢ 𝐴 ∈ On
2		onunisuc 6493	. 2 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∪ suc 𝐴 = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  ∪ cuni 4870  Oncon0 6380  suc csuc 6382

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-ss 3901  df-un 4154  df-sn 4589  df-pr 4591  df-uni 4871  df-tr 5216  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6383  df-on 6384  df-suc 6386

This theorem is used by:  rankuni  9906  onsucconni  37403  onsucsuccmpi  37409  finxp1o  38507