Theorem trsucss 4452

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 e. suc A -> B C_ A ) )

Proof of Theorem trsucss

Step	Hyp	Ref	Expression
1		elsuci 4432	. 2 |- ( B e. suc A -> ( B e. A \/ B = A ) )
2		trss 4136	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
3		eqimss 3233	. . . 4 |- ( B = A -> B C_ A )
4	3	a1i 9	. . 3 |- ( Tr A -> ( B = A -> B C_ A ) )
5	2, 4	jaod 718	. 2 |- ( Tr A -> ( ( B e. A \/ B = A ) -> B C_ A ) )
6	1, 5	syl5 32	1 |- ( Tr A -> ( B e. suc A -> B C_ A ) )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1364   e. wcel 2164   C_ wss 3153   Tr wtr 4127   suc csuc 4394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-uni 3836  df-tr 4128  df-suc 4400

This theorem is referenced by:  onsucsssucr  4539  ordpwsucss  4597  nnnninfeq  7181  bj-el2oss1o  15244  nnsf  15473