Theorem trsuc 2308

Description: A set whose successor belongs to a transitive class also belongs.

Assertion

Ref	Expression
trsuc	|- ((Tr A /\ suc B e. A) -> B e. A)

Proof of Theorem trsuc

Step	Hyp	Ref	Expression
1		trel 2048	. . . . . 6 |- (Tr A -> ((B e. suc B /\ suc B e. A) -> B e. A))
2	1	exp3a 292	. . . . 5 |- (Tr A -> (B e. suc B -> (suc B e. A -> B e. A)))
3		sucidg 2305	. . . . 5 |- (B e. V -> B e. suc B)
4	2, 3	syl5 22	. . . 4 |- (Tr A -> (B e. V -> (suc B e. A -> B e. A)))
5	4	com12 13	. . 3 |- (B e. V -> (Tr A -> (suc B e. A -> B e. A)))
6		sucprc 2297	. . . . . 6 |- (-. B e. V -> suc B = B)
7	6	eleq1d 1155	. . . . 5 |- (-. B e. V -> (suc B e. A <-> B e. A))
8	7	biimpd 135	. . . 4 |- (-. B e. V -> (suc B e. A -> B e. A))
9	8	a1d 14	. . 3 |- (-. B e. V -> (Tr A -> (suc B e. A -> B e. A)))
10	5, 9	pm2.61i 110	. 2 |- (Tr A -> (suc B e. A -> B e. A))
11	10	imp 277	1 |- ((Tr A /\ suc B e. A) -> B e. A)

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   e. wcel 1092  Vcvv 1348  Tr wtr 2041  suc csuc 2201

This theorem is referenced by:  onuninsuc 2356  limsuc 2361

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-uni 1920  df-tr 2042  df-suc 2205

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