Theorem ssorduni 4609

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 3918	. . . . 5 |- ( x e. U. A <-> E. y e. A x e. y )
2		ssel 3232	. . . . . . . . 9 |- ( A C_ On -> ( y e. A -> y e. On ) )
3		onelss 4508	. . . . . . . . 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 3936	. . . . . . 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 2659	. . . . 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 2614	. . 3 |- ( A C_ On -> A. x e. U. A x C_ U. A )
12		dftr3 4212	. . 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 4505	. . . . . . 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 2659	. . . 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 3244	. 2 |- ( A C_ On -> U. A C_ On )
20		ordon 4608	. . 3 |- Ord On
21		trssord 4501	. . . 4 |- ( ( Tr U. A /\ U. A C_ On /\ Ord On ) -> Ord U. A )
22	21	3exp 1229	. . 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 2203   A.wral 2520   E.wrex 2521   C_ wss 3211   U.cuni 3914   Tr wtr 4208   Ord word 4483   Oncon0 4484

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-in 3217  df-ss 3224  df-uni 3915  df-tr 4209  df-iord 4487  df-on 4489

This theorem is referenced by:  ssonuni  4610  orduni  4617  tfrlem8  6549  tfrexlem  6565