Theorem ssuniOLD 3202

Description: Subclass relationship for class union.

Assertion

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

Proof of Theorem ssuniOLD

Step	Hyp	Ref	Expression
1		sseq2 2639	. . . 4 |- (x = B -> (A C_ x <-> A C_ B))
2	1	imbi1d 675	. . 3 |- (x = B -> ((A C_ x -> A C_ U.C) <-> (A C_ B -> A C_ U.C)))
3		19.8a 1376	. . . . . . . 8 |- ((y e. x /\ x e. C) -> E.x(y e. x /\ x e. C))
4	3	expcom 403	. . . . . . 7 |- (x e. C -> (y e. x -> E.x(y e. x /\ x e. C)))
5		eluni 3180	. . . . . . 7 |- (y e. U.C <-> E.x(y e. x /\ x e. C))
6	4, 5	syl6ibr 230	. . . . . 6 |- (x e. C -> (y e. x -> y e. U.C))
7	6	imim2d 28	. . . . 5 |- (x e. C -> ((y e. A -> y e. x) -> (y e. A -> y e. U.C)))
8	7	alimdv 1668	. . . 4 |- (x e. C -> (A.y(y e. A -> y e. x) -> A.y(y e. A -> y e. U.C)))
9		dfss2 2610	. . . 4 |- (A C_ x <-> A.y(y e. A -> y e. x))
10		dfss2 2610	. . . 4 |- (A C_ U.C <-> A.y(y e. A -> y e. U.C))
11	8, 9, 10	3imtr4g 612	. . 3 |- (x e. C -> (A C_ x -> A C_ U.C))
12	2, 11	vtoclga 2352	. 2 |- (B e. C -> (A C_ B -> A C_ U.C))
13	12	impcom 378	1 |- ((A C_ B /\ B e. C) -> A C_ U.C)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326   C_ wss 2593  U.cuni 3177

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-ss 2605  df-uni 3178

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