Theorem uniss 3810

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 3136	. . . . 5 |- ( A C_ B -> ( y e. A -> y e. B ) )
2	1	anim2d 335	. . . 4 |- ( A C_ B -> ( ( x e. y /\ y e. A ) -> ( x e. y /\ y e. B ) ) )
3	2	eximdv 1868	. . 3 |- ( A C_ B -> ( E. y ( x e. y /\ y e. A ) -> E. y ( x e. y /\ y e. B ) ) )
4		eluni 3792	. . 3 |- ( x e. U. A <-> E. y ( x e. y /\ y e. A ) )
5		eluni 3792	. . 3 |- ( x e. U. B <-> E. y ( x e. y /\ y e. B ) )
6	3, 4, 5	3imtr4g 204	. 2 |- ( A C_ B -> ( x e. U. A -> x e. U. B ) )
7	6	ssrdv 3148	1 |- ( A C_ B -> U. A C_ U. B )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1480   e. wcel 2136   C_ wss 3116   U.cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790

This theorem is referenced by:  unissi  3812  unissd  3813  intssuni2m  3848  relfld  5132  tgcl  12704  distop  12725