Theorem trsuc 4968

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 4961	. . . . . 6 |- B C_ suc B
2		ssexg 4599	. . . . . 6 |- ( ( B C_ suc B /\ suc B e. A ) -> B e. _V )
3	1, 2	mpan 670	. . . . 5 |- ( suc B e. A -> B e. _V )
4		sucidg 4962	. . . . 5 |- ( B e. _V -> B e. suc B )
5	3, 4	syl 16	. . . 4 |- ( suc B e. A -> B e. suc B )
6	5	ancri 552	. . 3 |- ( suc B e. A -> ( B e. suc B /\ suc B e. A ) )
7		trel 4553	. . 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 429	1 |- ( ( Tr A /\ suc B e. A ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 369   e. wcel 1767   _Vcvv 3118   C_ wss 3481   Tr wtr 4546   suc csuc 4886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-un 3486  df-in 3488  df-ss 3495  df-sn 4034  df-uni 4252  df-tr 4547  df-suc 4890

This theorem is referenced by:  onuninsuci  6670  limsuc  6679  tz7.44-2  7085  cantnflt  8103  cantnfp1lem3  8111  cantnflem1b  8117  cantnflem1  8120  cantnfltOLD  8133  cantnfp1lem3OLD  8137  cantnflem1bOLD  8140  cantnflem1OLD  8143  cnfcom  8156  cnfcomOLD  8164  axdc3lem2  8843  inar1  9165  limsuc2  30914  bnj967  33483