Theorem trsucss 4083

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 4063	. 2 ⊢ (B ∈ suc A → (B ∈ A ∨ B = A))
2		trss 3815	. . 3 ⊢ (Tr A → (B ∈ A → B ⊆ A))
3		eqimss 2975	. . . 4 ⊢ (B = A → B ⊆ A)
4	3	a1i 9	. . 3 ⊢ (Tr A → (B = A → B ⊆ A))
5	2, 4	jaod 624	. 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 616   = wceq 1373   ∈ wcel 1375   ⊆ wss 2895  Tr wtr 3806  suc csuc 4026

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 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-sn 3333  df-uni 3533  df-tr 3807  df-suc 4031

This theorem is referenced by:  onsucsssucr  4157  ordpwsucss  4196