Theorem intss1 2548

Description: An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes.

Assertion

Ref	Expression
intss1	|- (A e. B -> |^|B (_ A)

Proof of Theorem intss1

Step	Hyp	Ref	Expression
1		eleq1 1534	. . . . . 6 |- (y = A -> (y e. B <-> A e. B))
2		eleq2 1535	. . . . . 6 |- (y = A -> (x e. y <-> x e. A))
3	1, 2	imbi12d 626	. . . . 5 |- (y = A -> ((y e. B -> x e. y) <-> (A e. B -> x e. A)))
4	3	cla4gv 1862	. . . 4 |- (A e. B -> (A.y(y e. B -> x e. y) -> (A e. B -> x e. A)))
5	4	pm2.43a 66	. . 3 |- (A e. B -> (A.y(y e. B -> x e. y) -> x e. A))
6		visset 1813	. . . 4 |- x e. V
7	6	elint 2539	. . 3 |- (x e. |^|B <-> A.y(y e. B -> x e. y))
8	5, 7	syl5ib 206	. 2 |- (A e. B -> (x e. |^|B -> x e. A))
9	8	ssrdv 2070	1 |- (A e. B -> |^|B (_ A)

Syntax hints:   -> wi 3  A.wal 954   = wceq 956   e. wcel 958   (_ wss 2047  |^|cint 2533

This theorem is referenced by:  intmin3 2558  intab 2560  int0el 2561  intex 2729  onint 3006  onssmin 3008  onintss 3011  onnmin 3015  oneqmini 3017  rankuni2 4690  cardonle 4822  peano5nn 5926  peano5uz 6203  shintcl 9293  ococint 9297  chsupsn 9312  shsumval2 9360

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-int 2534

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