Theorem trsuc 4914

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 4907	. . . . . 6 |- B C_ suc B
2		ssexg 4549	. . . . . 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 4908	. . . . 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 4503	. . 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 1758   _Vcvv 3078   C_ wss 3439   Tr wtr 4496   suc csuc 4832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-un 3444  df-in 3446  df-ss 3453  df-sn 3989  df-uni 4203  df-tr 4497  df-suc 4836

This theorem is referenced by:  onuninsuci  6564  limsuc  6573  tz7.44-2  6976  cantnflt  7995  cantnfp1lem3  8003  cantnflem1b  8009  cantnflem1  8012  cantnfltOLD  8025  cantnfp1lem3OLD  8029  cantnflem1bOLD  8032  cantnflem1OLD  8035  cnfcom  8048  cnfcomOLD  8056  axdc3lem2  8735  inar1  9057  limsuc2  29564  bnj967  32293