Theorem ordunisuc2r 4562

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 2775	. . . . . . . . 9 |- x e. _V
2	1	sucid 4464	. . . . . . . 8 |- x e. suc x
3		elunii 3855	. . . . . . . 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 1478	. . . . 5 |- ( A. x ( x e. A -> suc x e. A ) -> A. x ( x e. A -> x e. U. A ) )
7		df-ral 2489	. . . . 5 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
8		ssalel 3181	. . . . 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 4472	. . 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 3208	. 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 1371   = wceq 1373   e. wcel 2176   A.wral 2484   C_ wss 3166   U.cuni 3850   Ord word 4409   suc csuc 4412

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-uni 3851  df-tr 4143  df-iord 4413  df-suc 4418

This theorem is referenced by: (None)