Theorem trsuc 5810

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		trel 4759	. 2 |- ( Tr A -> ( ( B e. suc B /\ suc B e. A ) -> B e. A ) )
2		sssucid 5802	. . . . 5 |- B C_ suc B
3		ssexg 4804	. . . . 5 |- ( ( B C_ suc B /\ suc B e. A ) -> B e. _V )
4	2, 3	mpan 706	. . . 4 |- ( suc B e. A -> B e. _V )
5		sucidg 5803	. . . 4 |- ( B e. _V -> B e. suc B )
6	4, 5	syl 17	. . 3 |- ( suc B e. A -> B e. suc B )
7	6	ancri 575	. 2 |- ( suc B e. A -> ( B e. suc B /\ suc B e. A ) )
8	1, 7	impel 485	1 |- ( ( Tr A /\ suc B e. A ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   Tr wtr 4752   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-uni 4437  df-tr 4753  df-suc 5729

This theorem is referenced by:  onuninsuci  7040  limsuc  7049  tz7.44-2  7503  cantnflt  8569  cantnfp1lem3  8577  cantnflem1b  8583  cantnflem1  8586  cnfcom  8597  axdc3lem2  9273  inar1  9597  bnj967  31015  limsuc2  37611