Theorem onunisuci 4349

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 4344	. 2 |- Tr A
3	1	elexi 2693	. . 3 |- A e. _V
4	3	unisuc 4330	. 2 |- ( Tr A <-> U. suc A = A )
5	2, 4	mpbi 144	1 |- U. suc A = A

Syntax hints:   = wceq 1331   e. wcel 1480   U.cuni 3731   Tr wtr 4021   Oncon0 4280   suc csuc 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288

This theorem is referenced by: (None)