Theorem trsucss 4125

Description: A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)

Assertion

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

Proof of Theorem trsucss

Step	Hyp	Ref	Expression
1		elsuci 4105	. 2 ⊢ (B ∈ suc A → (B ∈ A ∨ B = A))
2		trss 3853	. . 3 ⊢ (Tr A → (B ∈ A → B ⊆ A))
3		eqimss 2991	. . . 4 ⊢ (B = A → B ⊆ A)
4	3	a1i 9	. . 3 ⊢ (Tr A → (B = A → B ⊆ A))
5	2, 4	jaod 636	. 2 ⊢ (Tr A → ((B ∈ A ∨ B = A) → B ⊆ A))
6	1, 5	syl5 28	1 ⊢ (Tr A → (B ∈ suc A → B ⊆ A))

Syntax hints:   → wi 4   ∨ wo 628   = wceq 1242   ∈ wcel 1390   ⊆ wss 2911  Tr wtr 3844  suc csuc 4067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3372  df-uni 3571  df-tr 3845  df-suc 4073

This theorem is referenced by:  onsucsssucr  4199  ordpwsucss  4242