Theorem trsucss 2309

Description: A member of the successor of a transitive class is a subclass of it.

Assertion

Ref	Expression
trsucss	⊢ (Tr A → (B ∈ suc A → B ⊆ A))

Proof of Theorem trsucss

Step	Hyp	Ref	Expression
1		trss 2050	. . 3 ⊢ (Tr A → (B ∈ A → B ⊆ A))
2		eqimss 1548	. . . 4 ⊢ (B = A → B ⊆ A)
3	2	a1i 7	. . 3 ⊢ (Tr A → (B = A → B ⊆ A))
4	1, 3	jaod 329	. 2 ⊢ (Tr A → ((B ∈ A ∨ B = A) → B ⊆ A))
5		elsuci 2289	. 2 ⊢ (B ∈ suc A → (B ∈ A ∨ B = A))
6	4, 5	syl5 22	1 ⊢ (Tr A → (B ∈ suc A → B ⊆ A))

Syntax hints:   → wi 2   ∨ wo 195   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  Tr wtr 2041  suc csuc 2201

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-or 197  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-un 1490  df-in 1491  df-ss 1492  df-sn 1811  df-pr 1812  df-uni 1920  df-tr 2042  df-suc 2205

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