Theorem onunisuci 6440

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 6431	. 2 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∪ suc 𝐴 = 𝐴

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∪ cuni 4851  Oncon0 6319  suc csuc 6321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3432  df-un 3895  df-ss 3907  df-sn 4569  df-pr 4571  df-uni 4852  df-tr 5194  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-ord 6322  df-on 6323  df-suc 6325

This theorem is referenced by:  rankuni  9782  onsucconni  36639  onsucsuccmpi  36645  finxp1o  37726