Theorem ssuni 3623

Description: Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
ssuni	|- ( ( A C_ B /\ B e. C ) -> A C_ U. C )

Proof of Theorem ssuni

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

Step	Hyp	Ref	Expression
1		eleq2 2142	. . . . . . 7 |- ( x = B -> ( y e. x <-> y e. B ) )
2	1	imbi1d 229	. . . . . 6 |- ( x = B -> ( ( y e. x -> y e. U. C ) <-> ( y e. B -> y e. U. C ) ) )
3		elunii 3606	. . . . . . 7 |- ( ( y e. x /\ x e. C ) -> y e. U. C )
4	3	expcom 114	. . . . . 6 |- ( x e. C -> ( y e. x -> y e. U. C ) )
5	2, 4	vtoclga 2664	. . . . 5 |- ( B e. C -> ( y e. B -> y e. U. C ) )
6	5	imim2d 53	. . . 4 |- ( B e. C -> ( ( y e. A -> y e. B ) -> ( y e. A -> y e. U. C ) ) )
7	6	alimdv 1800	. . 3 |- ( B e. C -> ( A. y ( y e. A -> y e. B ) -> A. y ( y e. A -> y e. U. C ) ) )
8		dfss2 2988	. . 3 |- ( A C_ B <-> A. y ( y e. A -> y e. B ) )
9		dfss2 2988	. . 3 |- ( A C_ U. C <-> A. y ( y e. A -> y e. U. C ) )
10	7, 8, 9	3imtr4g 203	. 2 |- ( B e. C -> ( A C_ B -> A C_ U. C ) )
11	10	impcom 123	1 |- ( ( A C_ B /\ B e. C ) -> A C_ U. C )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   = wceq 1284   e. wcel 1433   C_ wss 2973   U.cuni 3601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602

This theorem is referenced by:  elssuni  3629  uniss2  3632  ssorduni  4231