Theorem trss 2694

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 1537	. . . . 5 |- (x = B -> (x e. A <-> B e. A))
2		sseq1 2085	. . . . 5 |- (x = B -> (x (_ A <-> B (_ A))
3	1, 2	imbi12d 628	. . . 4 |- (x = B -> ((x e. A -> x (_ A) <-> (B e. A -> B (_ A)))
4	3	imbi2d 614	. . 3 |- (x = B -> ((Tr A -> (x e. A -> x (_ A)) <-> (Tr A -> (B e. A -> B (_ A))))
5		dftr3 2689	. . . 4 |- (Tr A <-> A.x e. A x (_ A)
6		ra4 1697	. . . 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 1850	. 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 958   e. wcel 960  A.wral 1648   (_ wss 2050  Tr wtr 2685

This theorem is referenced by:  trin 2695  tz7.2 2937  ordelss 2970  ordelord 2976  tz7.7 2979  onfr 2992  ssorduni 2999  onelsst 3006  trsucss 3062  r1tr 4664  r1ord 4665  r1ord2 4666

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-in 2054  df-ss 2056  df-uni 2508  df-tr 2686

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