Theorem onunisuci 4529

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 4524	. 2 |- Tr A
3	1	elexi 2815	. . 3 |- A e. _V
4	3	unisuc 4510	. 2 |- ( Tr A <-> U. suc A = A )
5	2, 4	mpbi 145	1 |- U. suc A = A

Syntax hints:   = wceq 1397   e. wcel 2202   U.cuni 3893   Tr wtr 4187   Oncon0 4460   suc csuc 4462

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468

This theorem is referenced by: (None)