Theorem ordunisuc2r 4546

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 2763	. . . . . . . . 9 |- x e. _V
2	1	sucid 4448	. . . . . . . 8 |- x e. suc x
3		elunii 3840	. . . . . . . 8 |- ( ( x e. suc x /\ suc x e. A ) -> x e. U. A )
4	2, 3	mpan 424	. . . . . . 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 1466	. . . . 5 |- ( A. x ( x e. A -> suc x e. A ) -> A. x ( x e. A -> x e. U. A ) )
7		df-ral 2477	. . . . 5 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
8		dfss2 3168	. . . . 5 |- ( A C_ U. A <-> A. x ( x e. A -> x e. U. A ) )
9	6, 7, 8	3imtr4i 201	. . . 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 4456	. . 3 |- ( Ord A -> U. A C_ A )
12	10, 11	jctird 317	. 2 |- ( Ord A -> ( A. x e. A suc x e. A -> ( A C_ U. A /\ U. A C_ A ) ) )
13		eqss 3194	. 2 |- ( A = U. A <-> ( A C_ U. A /\ U. A C_ A ) )
14	12, 13	imbitrrdi 162	1 |- ( Ord A -> ( A. x e. A suc x e. A -> A = U. A ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   e. wcel 2164   A.wral 2472   C_ wss 3153   U.cuni 3835   Ord word 4393   suc csuc 4396

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-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-uni 3836  df-tr 4128  df-iord 4397  df-suc 4402

This theorem is referenced by: (None)