Theorem trsuc 4971

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 4964	. . . . . 6 |- B C_ suc B
2		ssexg 4602	. . . . . 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 4965	. . . . 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 4557	. . 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 1819   _Vcvv 3109   C_ wss 3471   Tr wtr 4550   suc csuc 4889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3476  df-in 3478  df-ss 3485  df-sn 4033  df-uni 4252  df-tr 4551  df-suc 4893

This theorem is referenced by:  onuninsuci  6674  limsuc  6683  tz7.44-2  7091  cantnflt  8108  cantnfp1lem3  8116  cantnflem1b  8122  cantnflem1  8125  cantnfltOLD  8138  cantnfp1lem3OLD  8142  cantnflem1bOLD  8145  cantnflem1OLD  8148  cnfcom  8161  cnfcomOLD  8169  axdc3lem2  8848  inar1  9170  limsuc2  31148  bnj967  34104