Theorem ordunisuc2r 4240

Description: An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.)

Assertion

Ref	Expression
ordunisuc2r	|- ( Ord A -> ( A. x e. A suc x e. A -> A = U. A ) )

Distinct variable group:   x, A

Proof of Theorem ordunisuc2r

Step	Hyp	Ref	Expression
1		vex 2560	. . . . . . . . 9 |- x e. _V
2	1	sucid 4154	. . . . . . . 8 |- x e. suc x
3		elunii 3585	. . . . . . . 8 |- ( ( x e. suc x /\ suc x e. A ) -> x e. U. A )
4	2, 3	mpan 400	. . . . . . 7 |- ( suc x e. A -> x e. U. A )
5	4	imim2i 12	. . . . . 6 |- ( ( x e. A -> suc x e. A ) -> ( x e. A -> x e. U. A ) )
6	5	alimi 1344	. . . . 5 |- ( A. x ( x e. A -> suc x e. A ) -> A. x ( x e. A -> x e. U. A ) )
7		df-ral 2311	. . . . 5 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
8		dfss2 2934	. . . . 5 |- ( A C_ U. A <-> A. x ( x e. A -> x e. U. A ) )
9	6, 7, 8	3imtr4i 190	. . . 4 |- ( A. x e. A suc x e. A -> A C_ U. A )
10	9	a1i 9	. . 3 |- ( Ord A -> ( A. x e. A suc x e. A -> A C_ U. A ) )
11		orduniss 4162	. . 3 |- ( Ord A -> U. A C_ A )
12	10, 11	jctird 300	. 2 |- ( Ord A -> ( A. x e. A suc x e. A -> ( A C_ U. A /\ U. A C_ A ) ) )
13		eqss 2960	. 2 |- ( A = U. A <-> ( A C_ U. A /\ U. A C_ A ) )
14	12, 13	syl6ibr 151	1 |- ( Ord A -> ( A. x e. A suc x e. A -> A = U. A ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   e. wcel 1393   A.wral 2306   C_ wss 2917   U.cuni 3580   Ord word 4099   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-uni 3581  df-tr 3855  df-iord 4103  df-suc 4108

This theorem is referenced by: (None)