Theorem onunisuci 4461

Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)

Hypothesis

Ref	Expression
on.1	|- A e. On

Assertion

Ref	Expression
onunisuci	|- U. suc A = A

Proof of Theorem onunisuci

Step	Hyp	Ref	Expression
1		on.1	. . 3 |- A e. On
2	1	ontrci 4456	. 2 |- Tr A
3	1	elexi 2772	. . 3 |- A e. _V
4	3	unisuc 4442	. 2 |- ( Tr A <-> U. suc A = A )
5	2, 4	mpbi 145	1 |- U. suc A = A

Syntax hints:   = wceq 1364   e. wcel 2164   U.cuni 3835   Tr wtr 4127   Oncon0 4392   suc csuc 4394

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4395  df-on 4397  df-suc 4400

This theorem is referenced by: (None)