Theorem ssorduni 4579

Description: The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
ssorduni	|- ( A C_ On -> Ord U. A )

Proof of Theorem ssorduni

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni2 3892	. . . . 5 |- ( x e. U. A <-> E. y e. A x e. y )
2		ssel 3218	. . . . . . . . 9 |- ( A C_ On -> ( y e. A -> y e. On ) )
3		onelss 4478	. . . . . . . . 9 |- ( y e. On -> ( x e. y -> x C_ y ) )
4	2, 3	syl6 33	. . . . . . . 8 |- ( A C_ On -> ( y e. A -> ( x e. y -> x C_ y ) ) )
5		anc2r 328	. . . . . . . 8 |- ( ( y e. A -> ( x e. y -> x C_ y ) ) -> ( y e. A -> ( x e. y -> ( x C_ y /\ y e. A ) ) ) )
6	4, 5	syl 14	. . . . . . 7 |- ( A C_ On -> ( y e. A -> ( x e. y -> ( x C_ y /\ y e. A ) ) ) )
7		ssuni 3910	. . . . . . 7 |- ( ( x C_ y /\ y e. A ) -> x C_ U. A )
8	6, 7	syl8 71	. . . . . 6 |- ( A C_ On -> ( y e. A -> ( x e. y -> x C_ U. A ) ) )
9	8	rexlimdv 2647	. . . . 5 |- ( A C_ On -> ( E. y e. A x e. y -> x C_ U. A ) )
10	1, 9	biimtrid 152	. . . 4 |- ( A C_ On -> ( x e. U. A -> x C_ U. A ) )
11	10	ralrimiv 2602	. . 3 |- ( A C_ On -> A. x e. U. A x C_ U. A )
12		dftr3 4186	. . 3 |- ( Tr U. A <-> A. x e. U. A x C_ U. A )
13	11, 12	sylibr 134	. 2 |- ( A C_ On -> Tr U. A )
14		onelon 4475	. . . . . . 7 |- ( ( y e. On /\ x e. y ) -> x e. On )
15	14	ex 115	. . . . . 6 |- ( y e. On -> ( x e. y -> x e. On ) )
16	2, 15	syl6 33	. . . . 5 |- ( A C_ On -> ( y e. A -> ( x e. y -> x e. On ) ) )
17	16	rexlimdv 2647	. . . 4 |- ( A C_ On -> ( E. y e. A x e. y -> x e. On ) )
18	1, 17	biimtrid 152	. . 3 |- ( A C_ On -> ( x e. U. A -> x e. On ) )
19	18	ssrdv 3230	. 2 |- ( A C_ On -> U. A C_ On )
20		ordon 4578	. . 3 |- Ord On
21		trssord 4471	. . . 4 |- ( ( Tr U. A /\ U. A C_ On /\ Ord On ) -> Ord U. A )
22	21	3exp 1226	. . 3 |- ( Tr U. A -> ( U. A C_ On -> ( Ord On -> Ord U. A ) ) )
23	20, 22	mpii 44	. 2 |- ( Tr U. A -> ( U. A C_ On -> Ord U. A ) )
24	13, 19, 23	sylc 62	1 |- ( A C_ On -> Ord U. A )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   A.wral 2508   E.wrex 2509   C_ wss 3197   U.cuni 3888   Tr wtr 4182   Ord word 4453   Oncon0 4454

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-uni 3889  df-tr 4183  df-iord 4457  df-on 4459

This theorem is referenced by:  ssonuni  4580  orduni  4587  tfrlem8  6464  tfrexlem  6480