Theorem trsucss 3755

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

Assertion

Ref	Expression
trsucss	|- (Tr A -> (B e. suc A -> B C_ A))

Proof of Theorem trsucss

Step	Hyp	Ref	Expression
1		trss 3421	. . 3 |- (Tr A -> (B e. A -> B C_ A))
2		eqimss 2665	. . . 4 |- (B = A -> B C_ A)
3	2	a1i 8	. . 3 |- (Tr A -> (B = A -> B C_ A))
4	1, 3	jaod 469	. 2 |- (Tr A -> ((B e. A \/ B = A) -> B C_ A))
5		elsuci 3731	. 2 |- (B e. suc A -> (B e. A \/ B = A))
6	4, 5	syl5 20	1 |- (Tr A -> (B e. suc A -> B C_ A))

Syntax hints:   -> wi 3   \/ wo 239   = wceq 1298   e. wcel 1300   C_ wss 2593  Tr wtr 3411  suc csuc 3659

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-un 2600  df-in 2603  df-ss 2605  df-sn 3049  df-uni 3178  df-tr 3412  df-suc 3663

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