Theorem trel3 2693

Description: In a transitive class, the membership relation is transitive.

Assertion

Ref	Expression
trel3	|- (Tr A -> ((B e. C /\ C e. D /\ D e. A) -> B e. A))

Proof of Theorem trel3

Step	Hyp	Ref	Expression
1		trel 2692	. . . 4 |- (Tr A -> ((C e. D /\ D e. A) -> C e. A))
2	1	anim2d 563	. . 3 |- (Tr A -> ((B e. C /\ (C e. D /\ D e. A)) -> (B e. C /\ C e. A)))
3		3anass 781	. . 3 |- ((B e. C /\ C e. D /\ D e. A) <-> (B e. C /\ (C e. D /\ D e. A)))
4	2, 3	syl5ib 206	. 2 |- (Tr A -> ((B e. C /\ C e. D /\ D e. A) -> (B e. C /\ C e. A)))
5		trel 2692	. 2 |- (Tr A -> ((B e. C /\ C e. A) -> B e. A))
6	4, 5	syld 27	1 |- (Tr A -> ((B e. C /\ C e. D /\ D e. A) -> B e. A))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   e. wcel 960  Tr wtr 2685

This theorem is referenced by:  ordelord 2976

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-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-ss 2056  df-uni 2508  df-tr 2686

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