Theorem iunord 42422

Description: The indexed union of a collection of ordinal numbers B ( x ) is ordinal. This proof is based on the proof of ssorduni 6985, but does not use it directly, since ssorduni 6985 does not work when B is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.)

Assertion

Ref	Expression
iunord	|- ( A. x e. A Ord B -> Ord U_ x e. A B )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem iunord

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtr 5737	. . . 4 |- ( Ord B -> Tr B )
2	1	ralimi 2952	. . 3 |- ( A. x e. A Ord B -> A. x e. A Tr B )
3		triun 4766	. . 3 |- ( A. x e. A Tr B -> Tr U_ x e. A B )
4	2, 3	syl 17	. 2 |- ( A. x e. A Ord B -> Tr U_ x e. A B )
5		eliun 4524	. . . 4 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
6		nfra1 2941	. . . . 5 |- F/ x A. x e. A Ord B
7		nfv 1843	. . . . 5 |- F/ x y e. On
8		rsp 2929	. . . . . 6 |- ( A. x e. A Ord B -> ( x e. A -> Ord B ) )
9		ordelon 5747	. . . . . . 7 |- ( ( Ord B /\ y e. B ) -> y e. On )
10	9	ex 450	. . . . . 6 |- ( Ord B -> ( y e. B -> y e. On ) )
11	8, 10	syl6 35	. . . . 5 |- ( A. x e. A Ord B -> ( x e. A -> ( y e. B -> y e. On ) ) )
12	6, 7, 11	rexlimd 3026	. . . 4 |- ( A. x e. A Ord B -> ( E. x e. A y e. B -> y e. On ) )
13	5, 12	syl5bi 232	. . 3 |- ( A. x e. A Ord B -> ( y e. U_ x e. A B -> y e. On ) )
14	13	ssrdv 3609	. 2 |- ( A. x e. A Ord B -> U_ x e. A B C_ On )
15		ordon 6982	. . 3 |- Ord On
16		trssord 5740	. . . 4 |- ( ( Tr U_ x e. A B /\ U_ x e. A B C_ On /\ Ord On ) -> Ord U_ x e. A B )
17	16	3exp 1264	. . 3 |- ( Tr U_ x e. A B -> ( U_ x e. A B C_ On -> ( Ord On -> Ord U_ x e. A B ) ) )
18	15, 17	mpii 46	. 2 |- ( Tr U_ x e. A B -> ( U_ x e. A B C_ On -> Ord U_ x e. A B ) )
19	4, 14, 18	sylc 65	1 |- ( A. x e. A Ord B -> Ord U_ x e. A B )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   U_ciun 4520   Tr wtr 4752   Ord word 5722   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  iunordi  42423