Theorem ssuni 3871

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 2268	. . . . . . 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 3854	. . . . . . 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 2838	. . . . 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 1901	. . 3 |- ( B e. C -> ( A. y ( y e. A -> y e. B ) -> A. y ( y e. A -> y e. U. C ) ) )
8		ssalel 3180	. . 3 |- ( A C_ B <-> A. y ( y e. A -> y e. B ) )
9		ssalel 3180	. . 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 1370   = wceq 1372   e. wcel 2175   C_ wss 3165   U.cuni 3849

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-uni 3850

This theorem is referenced by:  elssuni  3877  uniss2  3880  ssorduni  4534