Theorem trsuc 4249

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 4242	. . . . . 6 |- B C_ suc B
2		ssexg 3978	. . . . . 6 |- ( ( B C_ suc B /\ suc B e. A ) -> B e. _V )
3	1, 2	mpan 415	. . . . 5 |- ( suc B e. A -> B e. _V )
4		sucidg 4243	. . . . 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 317	. . 3 |- ( suc B e. A -> ( B e. suc B /\ suc B e. A ) )
7		trel 3943	. . 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 122	1 |- ( ( Tr A /\ suc B e. A ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1438   _Vcvv 2619   C_ wss 2999   Tr wtr 3936   suc csuc 4192

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-uni 3654  df-tr 3937  df-suc 4198

This theorem is referenced by:  nnnninf  6806