Theorem ssuni 3766

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 2204	. . . . . . 7 |- ( x = B -> ( y e. x <-> y e. B ) )
2	1	imbi1d 230	. . . . . 6 |- ( x = B -> ( ( y e. x -> y e. U. C ) <-> ( y e. B -> y e. U. C ) ) )
3		elunii 3749	. . . . . . 7 |- ( ( y e. x /\ x e. C ) -> y e. U. C )
4	3	expcom 115	. . . . . 6 |- ( x e. C -> ( y e. x -> y e. U. C ) )
5	2, 4	vtoclga 2755	. . . . 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 1852	. . 3 |- ( B e. C -> ( A. y ( y e. A -> y e. B ) -> A. y ( y e. A -> y e. U. C ) ) )
8		dfss2 3091	. . 3 |- ( A C_ B <-> A. y ( y e. A -> y e. B ) )
9		dfss2 3091	. . 3 |- ( A C_ U. C <-> A. y ( y e. A -> y e. U. C ) )
10	7, 8, 9	3imtr4g 204	. 2 |- ( B e. C -> ( A C_ B -> A C_ U. C ) )
11	10	impcom 124	1 |- ( ( A C_ B /\ B e. C ) -> A C_ U. C )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1330   = wceq 1332   e. wcel 1481   C_ wss 3076   U.cuni 3744

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-uni 3745

This theorem is referenced by:  elssuni  3772  uniss2  3775  ssorduni  4411