Theorem trel3 2049

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

Assertion

Ref	Expression
trel3	⊢ (Tr A → ((B ∈ C ∧ C ∈ D ∧ D ∈ A) → B ∈ A))

Proof of Theorem trel3

Step	Hyp	Ref	Expression
1		trel 2048	. . . 4 ⊢ (Tr A → ((C ∈ D ∧ D ∈ A) → C ∈ A))
2	1	anim2d 433	. . 3 ⊢ (Tr A → ((B ∈ C ∧ (C ∈ D ∧ D ∈ A)) → (B ∈ C ∧ C ∈ A)))
3		3anass 585	. . 3 ⊢ ((B ∈ C ∧ C ∈ D ∧ D ∈ A) ↔ (B ∈ C ∧ (C ∈ D ∧ D ∈ A)))
4	2, 3	syl5ib 181	. 2 ⊢ (Tr A → ((B ∈ C ∧ C ∈ D ∧ D ∈ A) → (B ∈ C ∧ C ∈ A)))
5		trel 2048	. 2 ⊢ (Tr A → ((B ∈ C ∧ C ∈ A) → B ∈ A))
6	4, 5	syld 27	1 ⊢ (Tr A → ((B ∈ C ∧ C ∈ D ∧ D ∈ A) → B ∈ A))

Syntax hints:   → wi 2   ∧ wa 196   ∧ w3a 581   ∈ wcel 1092  Tr wtr 2041

This theorem is referenced by:  ordelord 2221

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920  df-tr 2042

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