Theorem ssuni 2522

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 2083	. . . 4 |- (x = B -> (A (_ x <-> A (_ B))
2	1	imbi1d 613	. . 3 |- (x = B -> ((A (_ x -> A (_ U.C) <-> (A (_ B -> A (_ U.C)))
3		19.8a 1029	. . . . . . . 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 2506	. . . . . . 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 1289	. . . 4 |- (x e. C -> (A.y(y e. A -> y e. x) -> A.y(y e. A -> y e. U.C)))
9		dfss2 2058	. . . 4 |- (A (_ x <-> A.y(y e. A -> y e. x))
10		dfss2 2058	. . . 4 |- (A (_ U.C <-> A.y(y e. A -> y e. U.C))
11	8, 9, 10	3imtr4g 553	. . 3 |- (x e. C -> (A (_ x -> A (_ U.C))
12	2, 11	vtoclga 1852	. 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 954   = wceq 956   e. wcel 958  E.wex 980   (_ wss 2047  U.cuni 2503

This theorem is referenced by:  elssuni 2526  uniss2 2529  ssorduni 2993  neiint 7719  opnuni 7868  fgsb 10570  fgsbOLD 10571  fgsb2 10580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-uni 2504

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