Theorem trsuc 4457

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 4450	. . . . . 6 |- B C_ suc B
2		ssexg 4172	. . . . . 6 |- ( ( B C_ suc B /\ suc B e. A ) -> B e. _V )
3	1, 2	mpan 424	. . . . 5 |- ( suc B e. A -> B e. _V )
4		sucidg 4451	. . . . 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 324	. . 3 |- ( suc B e. A -> ( B e. suc B /\ suc B e. A ) )
7		trel 4138	. . 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 124	1 |- ( ( Tr A /\ suc B e. A ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   _Vcvv 2763   C_ wss 3157   Tr wtr 4131   suc csuc 4400

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-sep 4151

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-uni 3840  df-tr 4132  df-suc 4406

This theorem is referenced by:  nnnninf  7192