Theorem trsuc 3752

Description: A set whose successor belongs to a transitive class also belongs. (The proof was 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 3418	. . 3 |- (Tr A -> ((B e. suc B /\ suc B e. A) -> B e. A))
2		sssucid 3742	. . . . . 6 |- B C_ suc B
3		ssexg 3457	. . . . . 6 |- ((B C_ suc B /\ suc B e. A) -> B e. _V)
4	2, 3	mpan 759	. . . . 5 |- (suc B e. A -> B e. _V)
5		sucidg 3743	. . . . 5 |- (B e. _V -> B e. suc B)
6	4, 5	syl 12	. . . 4 |- (suc B e. A -> B e. suc B)
7	6	ancri 321	. . 3 |- (suc B e. A -> (B e. suc B /\ suc B e. A))
8	1, 7	syl5 20	. 2 |- (Tr A -> (suc B e. A -> B e. A))
9	8	imp 377	1 |- ((Tr A /\ suc B e. A) -> B e. A)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  _Vcvv 2292   C_ wss 2593  Tr wtr 3411  suc csuc 3659

This theorem is referenced by:  onuninsuci 3921  limsuc 3933  bnj948 12847  ordsuccl 14430  ordsuccl2 14431  tartarmap 15265  smoge 16454

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-in 2603  df-ss 2605  df-sn 3049  df-uni 3178  df-tr 3412  df-suc 3663

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