Theorem trsuc 5526

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 5519	. . . . . 6 |- B C_ suc B
2		ssexg 4570	. . . . . 6 |- ( ( B C_ suc B /\ suc B e. A ) -> B e. _V )
3	1, 2	mpan 674	. . . . 5 |- ( suc B e. A -> B e. _V )
4		sucidg 5520	. . . . 5 |- ( B e. _V -> B e. suc B )
5	3, 4	syl 17	. . . 4 |- ( suc B e. A -> B e. suc B )
6	5	ancri 554	. . 3 |- ( suc B e. A -> ( B e. suc B /\ suc B e. A ) )
7		trel 4525	. . 3 |- ( Tr A -> ( ( B e. suc B /\ suc B e. A ) -> B e. A ) )
8	6, 7	syl5 33	. 2 |- ( Tr A -> ( suc B e. A -> B e. A ) )
9	8	imp 430	1 |- ( ( Tr A /\ suc B e. A ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 370   e. wcel 1872   _Vcvv 3080   C_ wss 3436   Tr wtr 4518   suc csuc 5444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-un 3441  df-in 3443  df-ss 3450  df-sn 3999  df-uni 4220  df-tr 4519  df-suc 5448

This theorem is referenced by:  onuninsuci  6682  limsuc  6691  tz7.44-2  7137  cantnflt  8186  cantnfp1lem3  8194  cantnflem1b  8200  cantnflem1  8203  cnfcom  8214  axdc3lem2  8889  inar1  9208  bnj967  29765  limsuc2  35870