Theorem ssuni 2512

Description: Subclass relationship for class union.

Assertion

Ref	Expression
ssuni	|- ((A (_ B /\ B e. C) -> A (_ U.C)

Proof of Theorem ssuni

Step	Hyp	Ref	Expression
1		sseq2 2073	. . . 4 |- (x = B -> (A (_ x <-> A (_ B))
2	1	imbi1d 611	. . 3 |- (x = B -> ((A (_ x -> A (_ U.C) <-> (A (_ B -> A (_ U.C)))
3		19.8a 1025	. . . . . . . 8 |- ((y e. x /\ x e. C) -> E.x(y e. x /\ x e. C))
4	3	expcom 374	. . . . . . 7 |- (x e. C -> (y e. x -> E.x(y e. x /\ x e. C)))
5		eluni 2496	. . . . . . 7 |- (y e. U.C <-> E.x(y e. x /\ x e. C))
6	4, 5	syl6ibr 213	. . . . . 6 |- (x e. C -> (y e. x -> y e. U.C))
7	6	imim2d 25	. . . . 5 |- (x e. C -> ((y e. A -> y e. x) -> (y e. A -> y e. U.C)))
8	7	19.20dv 1284	. . . 4 |- (x e. C -> (A.y(y e. A -> y e. x) -> A.y(y e. A -> y e. U.C)))
9		dfss2 2048	. . . 4 |- (A (_ x <-> A.y(y e. A -> y e. x))
10		dfss2 2048	. . . 4 |- (A (_ U.C <-> A.y(y e. A -> y e. U.C))
11	8, 9, 10	3imtr4g 551	. . 3 |- (x e. C -> (A (_ x -> A (_ U.C))
12	2, 11	vtoclga 1843	. 2 |- (B e. C -> (A (_ B -> A (_ U.C))
13	12	impcom 351	1 |- ((A (_ B /\ B e. C) -> A (_ U.C)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977   (_ wss 2037  U.cuni 2493

This theorem is referenced by:  elssuni 2516  uniss2 2519  ssorduni 2983  neiint 7660  opnuni 7808  fgsb 10444  fgsb2 10449

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-in 2041  df-ss 2043  df-uni 2494

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