Theorem onunisuci 4555

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 4550	. 2 |- Tr A
3	1	elexi 2828	. . 3 |- A e. _V
4	3	unisuc 4536	. 2 |- ( Tr A <-> U. suc A = A )
5	2, 4	mpbi 145	1 |- U. suc A = A

Syntax hints:   = wceq 1398   e. wcel 2205   U.cuni 3916   Tr wtr 4210   Oncon0 4486   suc csuc 4488

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-sn 3697  df-pr 3698  df-uni 3917  df-tr 4211  df-iord 4489  df-on 4491  df-suc 4494

This theorem is referenced by: (None)