Theorem ssuni 1937

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 1522	. . . . 5 |- (x = B -> (A (_ x <-> A (_ B))
2	1	imbi1d 465	. . . 4 |- (x = B -> ((A (_ x -> A (_ U.C) <-> (A (_ B -> A (_ U.C)))
3		19.8a 712	. . . . . . . . . 10 |- ((y e. x /\ x e. C) -> E.x(y e. x /\ x e. C))
4	3	exp 291	. . . . . . . . 9 |- (y e. x -> (x e. C -> E.x(y e. x /\ x e. C)))
5	4	com12 13	. . . . . . . 8 |- (x e. C -> (y e. x -> E.x(y e. x /\ x e. C)))
6		eluni 1922	. . . . . . . 8 |- (y e. U.C <-> E.x(y e. x /\ x e. C))
7	5, 6	syl6ibr 186	. . . . . . 7 |- (x e. C -> (y e. x -> y e. U.C))
8	7	syl3d 26	. . . . . 6 |- (x e. C -> ((y e. A -> y e. x) -> (y e. A -> y e. U.C)))
9	8	19.20dv 946	. . . . 5 |- (x e. C -> (A.y(y e. A -> y e. x) -> A.y(y e. A -> y e. U.C)))
10		dfss2 1497	. . . . 5 |- (A (_ x <-> A.y(y e. A -> y e. x))
11		dfss2 1497	. . . . 5 |- (A (_ U.C <-> A.y(y e. A -> y e. U.C))
12	9, 10, 11	3imtr4g 426	. . . 4 |- (x e. C -> (A (_ x -> A (_ U.C))
13	2, 12	vtoclga 1387	. . 3 |- (B e. C -> (A (_ B -> A (_ U.C))
14	13	com12 13	. 2 |- (A (_ B -> (B e. C -> A (_ U.C))
15	14	imp 277	1 |- ((A (_ B /\ B e. C) -> A (_ U.C)

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   e. wel 803   = wceq 1091   e. wcel 1092   (_ wss 1487  U.cuni 1919

This theorem is referenced by:  elssuni 1940  uniss2 1942  ssorduni 2249

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920

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