Theorem uniss 3951

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 3242	. . . . 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 1933	. . 3 |- ( A C_ B -> ( E. y ( x e. y /\ y e. A ) -> E. y ( x e. y /\ y e. B ) ) )
4		eluni 3933	. . 3 |- ( x e. U. A <-> E. y ( x e. y /\ y e. A ) )
5		eluni 3933	. . 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 3254	1 |- ( A C_ B -> U. A C_ U. B )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1545   e. wcel 2209   C_ wss 3220   U.cuni 3930

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-uni 3931

This theorem is referenced by:  unissi  3953  unissd  3954  intssuni2m  3989  relfld  5311  prdsvallem  13598  prdsval  14150  tgcl  15088  distop  15109