Theorem trsuc 4352

Description: A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
trsuc	|- ( ( Tr A /\ suc B e. A ) -> B e. A )

Proof of Theorem trsuc

Step	Hyp	Ref	Expression
1		sssucid 4345	. . . . . 6 |- B C_ suc B
2		ssexg 4075	. . . . . 6 |- ( ( B C_ suc B /\ suc B e. A ) -> B e. _V )
3	1, 2	mpan 421	. . . . 5 |- ( suc B e. A -> B e. _V )
4		sucidg 4346	. . . . 5 |- ( B e. _V -> B e. suc B )
5	3, 4	syl 14	. . . 4 |- ( suc B e. A -> B e. suc B )
6	5	ancri 322	. . 3 |- ( suc B e. A -> ( B e. suc B /\ suc B e. A ) )
7		trel 4041	. . 3 |- ( Tr A -> ( ( B e. suc B /\ suc B e. A ) -> B e. A ) )
8	6, 7	syl5 32	. 2 |- ( Tr A -> ( suc B e. A -> B e. A ) )
9	8	imp 123	1 |- ( ( Tr A /\ suc B e. A ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1481   _Vcvv 2689   C_ wss 3076   Tr wtr 4034   suc csuc 4295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-uni 3745  df-tr 4035  df-suc 4301

This theorem is referenced by:  nnnninf  7031