Theorem onunisuci 4535

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 4530	. 2 ⊢ Tr 𝐴
3	1	elexi 2816	. . 3 ⊢ 𝐴 ∈ V
4	3	unisuc 4516	. 2 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)
5	2, 4	mpbi 145	1 ⊢ ∪ suc 𝐴 = 𝐴

Syntax hints:   = wceq 1398   ∈ wcel 2202  ∪ cuni 3898  Tr wtr 4192  Oncon0 4466  suc csuc 4468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-uni 3899  df-tr 4193  df-iord 4469  df-on 4471  df-suc 4474

This theorem is referenced by: (None)