Theorem trss 2689

Description: An element of a transitive class is a subset of the class.

Assertion

Ref	Expression
trss	|- (Tr A -> (B e. A -> B (_ A))

Proof of Theorem trss

Step	Hyp	Ref	Expression
1		eleq1 1534	. . . . 5 |- (x = B -> (x e. A <-> B e. A))
2		sseq1 2082	. . . . 5 |- (x = B -> (x (_ A <-> B (_ A))
3	1, 2	imbi12d 626	. . . 4 |- (x = B -> ((x e. A -> x (_ A) <-> (B e. A -> B (_ A)))
4	3	imbi2d 612	. . 3 |- (x = B -> ((Tr A -> (x e. A -> x (_ A)) <-> (Tr A -> (B e. A -> B (_ A))))
5		dftr3 2684	. . . 4 |- (Tr A <-> A.x e. A x (_ A)
6		ra4 1694	. . . 4 |- (A.x e. A x (_ A -> (x e. A -> x (_ A))
7	5, 6	sylbi 199	. . 3 |- (Tr A -> (x e. A -> x (_ A))
8	4, 7	vtoclg 1847	. 2 |- (B e. A -> (Tr A -> (B e. A -> B (_ A)))
9	8	pm2.43b 67	1 |- (Tr A -> (B e. A -> B (_ A))

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047  Tr wtr 2680

This theorem is referenced by:  trin 2690  tz7.2 2931  ordelss 2964  ordelord 2970  tz7.7 2973  onfr 2986  ssorduni 2993  onelsst 3000  trsucss 3056  r1tr 4654  r1ord 4655  r1ord2 4656

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-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-uni 2504  df-tr 2681

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