Theorem ssuni 2517

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 2079	. . . 4 |- (x = B -> (A (_ x <-> A (_ B))
2	1	imbi1d 612	. . 3 |- (x = B -> ((A (_ x -> A (_ U.C) <-> (A (_ B -> A (_ U.C)))
3		19.8a 1027	. . . . . . . 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 2501	. . . . . . 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 1287	. . . 4 |- (x e. C -> (A.y(y e. A -> y e. x) -> A.y(y e. A -> y e. U.C)))
9		dfss2 2054	. . . 4 |- (A (_ x <-> A.y(y e. A -> y e. x))
10		dfss2 2054	. . . 4 |- (A (_ U.C <-> A.y(y e. A -> y e. U.C))
11	8, 9, 10	3imtr4g 552	. . 3 |- (x e. C -> (A (_ x -> A (_ U.C))
12	2, 11	vtoclga 1848	. 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 952   = wceq 954   e. wcel 956  E.wex 978   (_ wss 2043  U.cuni 2498

This theorem is referenced by:  elssuni 2521  uniss2 2524  ssorduni 2988  neiint 7669  opnuni 7820  fgsb 10480  fgsb2 10485

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-in 2047  df-ss 2049  df-uni 2499

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