Theorem ordunisuc2r 4258

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 2604	. . . . . . . . 9 |- x e. _V
2	1	sucid 4172	. . . . . . . 8 |- x e. suc x
3		elunii 3606	. . . . . . . 8 |- ( ( x e. suc x /\ suc x e. A ) -> x e. U. A )
4	2, 3	mpan 414	. . . . . . 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 1384	. . . . 5 |- ( A. x ( x e. A -> suc x e. A ) -> A. x ( x e. A -> x e. U. A ) )
7		df-ral 2353	. . . . 5 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
8		dfss2 2988	. . . . 5 |- ( A C_ U. A <-> A. x ( x e. A -> x e. U. A ) )
9	6, 7, 8	3imtr4i 199	. . . 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 4180	. . 3 |- ( Ord A -> U. A C_ A )
12	10, 11	jctird 310	. 2 |- ( Ord A -> ( A. x e. A suc x e. A -> ( A C_ U. A /\ U. A C_ A ) ) )
13		eqss 3014	. 2 |- ( A = U. A <-> ( A C_ U. A /\ U. A C_ A ) )
14	12, 13	syl6ibr 160	1 |- ( Ord A -> ( A. x e. A suc x e. A -> A = U. A ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   = wceq 1284   e. wcel 1433   A.wral 2348   C_ wss 2973   U.cuni 3601   Ord word 4117   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-uni 3602  df-tr 3876  df-iord 4121  df-suc 4126

This theorem is referenced by: (None)