Theorem trel 2048

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

Assertion

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

Proof of Theorem trel

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . . 6 ⊢ (x = B → (x ∈ C ↔ B ∈ C))
2		eleq1 1149	. . . . . . 7 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
3	2	imbi2d 464	. . . . . 6 ⊢ (x = B → ((C ∈ A → x ∈ A) ↔ (C ∈ A → B ∈ A)))
4	1, 3	imbi12d 474	. . . . 5 ⊢ (x = B → ((x ∈ C → (C ∈ A → x ∈ A)) ↔ (B ∈ C → (C ∈ A → B ∈ A))))
5	4	imbi2d 464	. . . 4 ⊢ (x = B → ((Tr A → (x ∈ C → (C ∈ A → x ∈ A))) ↔ (Tr A → (B ∈ C → (C ∈ A → B ∈ A)))))
6		eleq2 1150	. . . . . . . . 9 ⊢ (y = C → (x ∈ y ↔ x ∈ C))
7		eleq1 1149	. . . . . . . . . 10 ⊢ (y = C → (y ∈ A ↔ C ∈ A))
8	7	imbi1d 465	. . . . . . . . 9 ⊢ (y = C → ((y ∈ A → x ∈ A) ↔ (C ∈ A → x ∈ A)))
9	6, 8	imbi12d 474	. . . . . . . 8 ⊢ (y = C → ((x ∈ y → (y ∈ A → x ∈ A)) ↔ (x ∈ C → (C ∈ A → x ∈ A))))
10	9	imbi2d 464	. . . . . . 7 ⊢ (y = C → ((Tr A → (x ∈ y → (y ∈ A → x ∈ A))) ↔ (Tr A → (x ∈ C → (C ∈ A → x ∈ A)))))
11		dftr2 2043	. . . . . . . . . 10 ⊢ (Tr A ↔ ∀x∀y((x ∈ y ∧ y ∈ A) → x ∈ A))
12	11	biimp 133	. . . . . . . . 9 ⊢ (Tr A → ∀x∀y((x ∈ y ∧ y ∈ A) → x ∈ A))
13	12	19.21bbi 743	. . . . . . . 8 ⊢ (Tr A → ((x ∈ y ∧ y ∈ A) → x ∈ A))
14	13	exp3a 292	. . . . . . 7 ⊢ (Tr A → (x ∈ y → (y ∈ A → x ∈ A)))
15	10, 14	vtoclg 1383	. . . . . 6 ⊢ (C ∈ A → (Tr A → (x ∈ C → (C ∈ A → x ∈ A))))
16	15	com4l 39	. . . . 5 ⊢ (Tr A → (x ∈ C → (C ∈ A → (C ∈ A → x ∈ A))))
17		pm2.43 57	. . . . 5 ⊢ ((C ∈ A → (C ∈ A → x ∈ A)) → (C ∈ A → x ∈ A))
18	16, 17	syl6 23	. . . 4 ⊢ (Tr A → (x ∈ C → (C ∈ A → x ∈ A)))
19	5, 18	vtoclg 1383	. . 3 ⊢ (B ∈ C → (Tr A → (B ∈ C → (C ∈ A → B ∈ A))))
20	19	pm2.43b 61	. 2 ⊢ (Tr A → (B ∈ C → (C ∈ A → B ∈ A)))
21	20	imp3a 279	1 ⊢ (Tr A → ((B ∈ C ∧ C ∈ A) → B ∈ A))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   ∈ wel 803   = wceq 1091   ∈ wcel 1092  Tr wtr 2041

This theorem is referenced by:  trel3 2049  ordn2lp 2219  ordelord 2221  tz7.7 2224  ordtr1 2256  trsuc 2308  ordom 2382  elnn 2383  zfregs 3491

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-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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