Theorem uniss 3871

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 3187	. . . . 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 1903	. . 3 |- ( A C_ B -> ( E. y ( x e. y /\ y e. A ) -> E. y ( x e. y /\ y e. B ) ) )
4		eluni 3853	. . 3 |- ( x e. U. A <-> E. y ( x e. y /\ y e. A ) )
5		eluni 3853	. . 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 3199	1 |- ( A C_ B -> U. A C_ U. B )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1515   e. wcel 2176   C_ wss 3166   U.cuni 3850

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851

This theorem is referenced by:  unissi  3873  unissd  3874  intssuni2m  3909  relfld  5211  prdsvallem  13104  prdsval  13105  tgcl  14536  distop  14557