Theorem ssuni 3914

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 2414	. . . . . . 7 |- (x = B -> (y e. x <-> y e. B))
2	1	imbi1d 308	. . . . . 6 |- (x = B -> ((y e. x -> y e. U.C) <-> (y e. B -> y e. U.C)))
3		elunii 3897	. . . . . . 7 |- ((y e. x /\ x e. C) -> y e. U.C)
4	3	expcom 424	. . . . . 6 |- (x e. C -> (y e. x -> y e. U.C))
5	2, 4	vtoclga 2921	. . . . 5 |- (B e. C -> (y e. B -> y e. U.C))
6	5	imim2d 48	. . . 4 |- (B e. C -> ((y e. A -> y e. B) -> (y e. A -> y e. U.C)))
7	6	alimdv 1621	. . 3 |- (B e. C -> (A.y(y e. A -> y e. B) -> A.y(y e. A -> y e. U.C)))
8		dfss2 3263	. . 3 |- (A C_ B <-> A.y(y e. A -> y e. B))
9		dfss2 3263	. . 3 |- (A C_ U.C <-> A.y(y e. A -> y e. U.C))
10	7, 8, 9	3imtr4g 261	. 2 |- (B e. C -> (A C_ B -> A C_ U.C))
11	10	impcom 419	1 |- ((A C_ B /\ B e. C) -> A C_ U.C)

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710   C_ wss 3258  U.cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-uni 3893

This theorem is referenced by:  elssuni  3920  uniss2  3923