Theorem ordunisuc2r 4467

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 2712	. . . . . . . . 9 |- x e. _V
2	1	sucid 4372	. . . . . . . 8 |- x e. suc x
3		elunii 3773	. . . . . . . 8 |- ( ( x e. suc x /\ suc x e. A ) -> x e. U. A )
4	2, 3	mpan 421	. . . . . . 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 1432	. . . . 5 |- ( A. x ( x e. A -> suc x e. A ) -> A. x ( x e. A -> x e. U. A ) )
7		df-ral 2437	. . . . 5 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
8		dfss2 3113	. . . . 5 |- ( A C_ U. A <-> A. x ( x e. A -> x e. U. A ) )
9	6, 7, 8	3imtr4i 200	. . . 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 4380	. . 3 |- ( Ord A -> U. A C_ A )
12	10, 11	jctird 315	. 2 |- ( Ord A -> ( A. x e. A suc x e. A -> ( A C_ U. A /\ U. A C_ A ) ) )
13		eqss 3139	. 2 |- ( A = U. A <-> ( A C_ U. A /\ U. A C_ A ) )
14	12, 13	syl6ibr 161	1 |- ( Ord A -> ( A. x e. A suc x e. A -> A = U. A ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1330   = wceq 1332   e. wcel 2125   A.wral 2432   C_ wss 3098   U.cuni 3768   Ord word 4317   suc csuc 4320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-sn 3562  df-uni 3769  df-tr 4059  df-iord 4321  df-suc 4326

This theorem is referenced by: (None)