Theorem trsuc 5530

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 5523	. . . . . 6 |- B C_ suc B
2		ssexg 4565	. . . . . 6 |- ( ( B C_ suc B /\ suc B e. A ) -> B e. _V )
3	1, 2	mpan 681	. . . . 5 |- ( suc B e. A -> B e. _V )
4		sucidg 5524	. . . . 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 559	. . 3 |- ( suc B e. A -> ( B e. suc B /\ suc B e. A ) )
7		trel 4520	. . 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 435	1 |- ( ( Tr A /\ suc B e. A ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 375   e. wcel 1898   _Vcvv 3057   C_ wss 3416   Tr wtr 4513   suc csuc 5448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-un 3421  df-in 3423  df-ss 3430  df-sn 3981  df-uni 4213  df-tr 4514  df-suc 5452

This theorem is referenced by:  onuninsuci  6699  limsuc  6708  tz7.44-2  7156  cantnflt  8208  cantnfp1lem3  8216  cantnflem1b  8222  cantnflem1  8225  cnfcom  8236  axdc3lem2  8912  inar1  9231  bnj967  29806  limsuc2  35945