Theorem uniss 4458

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 3597	. . . . 5 |- ( A C_ B -> ( y e. A -> y e. B ) )
2	1	anim2d 589	. . . 4 |- ( A C_ B -> ( ( x e. y /\ y e. A ) -> ( x e. y /\ y e. B ) ) )
3	2	eximdv 1846	. . 3 |- ( A C_ B -> ( E. y ( x e. y /\ y e. A ) -> E. y ( x e. y /\ y e. B ) ) )
4		eluni 4439	. . 3 |- ( x e. U. A <-> E. y ( x e. y /\ y e. A ) )
5		eluni 4439	. . 3 |- ( x e. U. B <-> E. y ( x e. y /\ y e. B ) )
6	3, 4, 5	3imtr4g 285	. 2 |- ( A C_ B -> ( x e. U. A -> x e. U. B ) )
7	6	ssrdv 3609	1 |- ( A C_ B -> U. A C_ U. B )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   e. wcel 1990   C_ wss 3574   U.cuni 4436

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437

This theorem is referenced by:  unissi  4461  unissd  4462  intssuni2  4502  uniintsn  4514  relfld  5661  dffv2  6271  trcl  8604  cflm  9072  coflim  9083  cfslbn  9089  fin23lem41  9174  fin1a2lem12  9233  tskuni  9605  prdsval  16115  prdsbas  16117  prdsplusg  16118  prdsmulr  16119  prdsvsca  16120  prdshom  16127  mrcssv  16274  catcfuccl  16759  catcxpccl  16847  mrelatlub  17186  mreclatBAD  17187  dprdres  18427  dmdprdsplit2lem  18444  tgcl  20773  distop  20799  fctop  20808  cctop  20810  neiptoptop  20935  cmpcld  21205  uncmp  21206  cmpfi  21211  comppfsc  21335  kgentopon  21341  txcmplem2  21445  filconn  21687  alexsubALTlem3  21853  alexsubALT  21855  ptcmplem3  21858  dyadmbllem  23367  shsupcl  28197  hsupss  28200  shatomistici  29220  pwuniss  29370  carsggect  30380  carsgclctun  30383  cvmliftlem15  31280  frrlem5c  31786  filnetlem3  32375  icoreunrn  33207  heiborlem1  33610  lssats  34299  lpssat  34300  lssatle  34302  lssat  34303  dicval  36465  pwsal  40535  prsal  40538  intsaluni  40547