Theorem uniss 3908

Description: Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
uniss	|- ( A C_ B -> U. A C_ U. B )

Proof of Theorem uniss

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

Step	Hyp	Ref	Expression
1		ssel 3218	. . . . 5 |- ( A C_ B -> ( y e. A -> y e. B ) )
2	1	anim2d 337	. . . 4 |- ( A C_ B -> ( ( x e. y /\ y e. A ) -> ( x e. y /\ y e. B ) ) )
3	2	eximdv 1926	. . 3 |- ( A C_ B -> ( E. y ( x e. y /\ y e. A ) -> E. y ( x e. y /\ y e. B ) ) )
4		eluni 3890	. . 3 |- ( x e. U. A <-> E. y ( x e. y /\ y e. A ) )
5		eluni 3890	. . 3 |- ( x e. U. B <-> E. y ( x e. y /\ y e. B ) )
6	3, 4, 5	3imtr4g 205	. 2 |- ( A C_ B -> ( x e. U. A -> x e. U. B ) )
7	6	ssrdv 3230	1 |- ( A C_ B -> U. A C_ U. B )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1538   e. wcel 2200   C_ wss 3197   U.cuni 3887

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-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-uni 3888

This theorem is referenced by:  unissi  3910  unissd  3911  intssuni2m  3946  relfld  5256  prdsvallem  13300  prdsval  13301  tgcl  14732  distop  14753