Theorem ssuni 3920

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 2295	. . . . . . 7 |- ( x = B -> ( y e. x <-> y e. B ) )
2	1	imbi1d 231	. . . . . 6 |- ( x = B -> ( ( y e. x -> y e. U. C ) <-> ( y e. B -> y e. U. C ) ) )
3		elunii 3903	. . . . . . 7 |- ( ( y e. x /\ x e. C ) -> y e. U. C )
4	3	expcom 116	. . . . . 6 |- ( x e. C -> ( y e. x -> y e. U. C ) )
5	2, 4	vtoclga 2871	. . . . 5 |- ( B e. C -> ( y e. B -> y e. U. C ) )
6	5	imim2d 54	. . . 4 |- ( B e. C -> ( ( y e. A -> y e. B ) -> ( y e. A -> y e. U. C ) ) )
7	6	alimdv 1927	. . 3 |- ( B e. C -> ( A. y ( y e. A -> y e. B ) -> A. y ( y e. A -> y e. U. C ) ) )
8		ssalel 3216	. . 3 |- ( A C_ B <-> A. y ( y e. A -> y e. B ) )
9		ssalel 3216	. . 3 |- ( A C_ U. C <-> A. y ( y e. A -> y e. U. C ) )
10	7, 8, 9	3imtr4g 205	. 2 |- ( B e. C -> ( A C_ B -> A C_ U. C ) )
11	10	impcom 125	1 |- ( ( A C_ B /\ B e. C ) -> A C_ U. C )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1396   = wceq 1398   e. wcel 2202   C_ wss 3201   U.cuni 3898

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-uni 3899

This theorem is referenced by:  elssuni  3926  uniss2  3929  ssorduni  4591