Theorem trss 2050

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

Assertion

Ref	Expression
trss	⊢ (Tr A → (B ∈ A → B ⊆ A))

Proof of Theorem trss

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 5 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
2		sseq1 1521	. . . . 5 ⊢ (x = B → (x ⊆ A ↔ B ⊆ A))
3	1, 2	imbi12d 474	. . . 4 ⊢ (x = B → ((x ∈ A → x ⊆ A) ↔ (B ∈ A → B ⊆ A)))
4	3	imbi2d 464	. . 3 ⊢ (x = B → ((Tr A → (x ∈ A → x ⊆ A)) ↔ (Tr A → (B ∈ A → B ⊆ A))))
5		dftr3 2045	. . . 4 ⊢ (Tr A ↔ ∀x ∈ A x ⊆ A)
6		ra4 1243	. . . 4 ⊢ (∀x ∈ A x ⊆ A → (x ∈ A → x ⊆ A))
7	5, 6	sylbi 174	. . 3 ⊢ (Tr A → (x ∈ A → x ⊆ A))
8	4, 7	vtoclg 1383	. 2 ⊢ (B ∈ A → (Tr A → (B ∈ A → B ⊆ A)))
9	8	pm2.43b 61	1 ⊢ (Tr A → (B ∈ A → B ⊆ A))

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  Tr wtr 2041

This theorem is referenced by:  trin 2051  tz7.2 2183  ordelss 2215  ordelord 2221  tz7.7 2224  onfr 2237  ssorduni 2249  onelsst 2255  trsucss 2309  r1tr 3498  r1ord 3499  r1ord2 3500

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-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920  df-tr 2042

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