Theorem trsucss 4108

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 4087	. 2 ⊢ (B ∈ suc A → (B ∈ A ∨ B = A))
2		trss 3835	. . 3 ⊢ (Tr A → (B ∈ A → B ⊆ A))
3		eqimss 2972	. . . 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 1228   ∈ wcel 1374   ⊆ wss 2892  Tr wtr 3826  suc csuc 4049

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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-v 2535  df-un 2897  df-in 2899  df-ss 2906  df-sn 3354  df-uni 3553  df-tr 3827  df-suc 4055

This theorem is referenced by:  onsucsssucr  4182  ordpwsucss  4225