Theorem trsuc2 3754

Description: The successor of a transitive set is transitive. (Contributed by Scott Fenton, 21-Feb-2011.)

Assertion

Ref	Expression
trsuc2	|- (Tr A -> Tr suc A)

Proof of Theorem trsuc2

Step	Hyp	Ref	Expression
1		trel 3418	. . . . . . . . 9 |- (Tr A -> ((x e. y /\ y e. A) -> x e. A))
2		orc 291	. . . . . . . . 9 |- (x e. A -> (x e. A \/ x = A))
3	1, 2	syl6 25	. . . . . . . 8 |- (Tr A -> ((x e. y /\ y e. A) -> (x e. A \/ x = A)))
4	2	a1i 8	. . . . . . . 8 |- (Tr A -> (x e. A -> (x e. A \/ x = A)))
5	3, 4	jaod 469	. . . . . . 7 |- (Tr A -> (((x e. y /\ y e. A) \/ x e. A) -> (x e. A \/ x = A)))
6		eleq2 1958	. . . . . . . . 9 |- (y = A -> (x e. y <-> x e. A))
7	6	biimpac 462	. . . . . . . 8 |- ((x e. y /\ y = A) -> x e. A)
8	7	orim2i 365	. . . . . . 7 |- (((x e. y /\ y e. A) \/ (x e. y /\ y = A)) -> ((x e. y /\ y e. A) \/ x e. A))
9	5, 8	syl5 20	. . . . . 6 |- (Tr A -> (((x e. y /\ y e. A) \/ (x e. y /\ y = A)) -> (x e. A \/ x = A)))
10		elsn 3058	. . . . . . 7 |- (x e. {A} <-> x = A)
11	10	orbi2i 275	. . . . . 6 |- ((x e. A \/ x e. {A}) <-> (x e. A \/ x = A))
12	9, 11	syl6ibr 230	. . . . 5 |- (Tr A -> (((x e. y /\ y e. A) \/ (x e. y /\ y = A)) -> (x e. A \/ x e. {A})))
13		elsn 3058	. . . . . . 7 |- (y e. {A} <-> y = A)
14	13	anbi2i 538	. . . . . 6 |- ((x e. y /\ y e. {A}) <-> (x e. y /\ y = A))
15	14	orbi2i 275	. . . . 5 |- (((x e. y /\ y e. A) \/ (x e. y /\ y e. {A})) <-> ((x e. y /\ y e. A) \/ (x e. y /\ y = A)))
16	12, 15	syl5ib 223	. . . 4 |- (Tr A -> (((x e. y /\ y e. A) \/ (x e. y /\ y e. {A})) -> (x e. A \/ x e. {A})))
17		andi 665	. . . 4 |- ((x e. y /\ (y e. A \/ y e. {A})) <-> ((x e. y /\ y e. A) \/ (x e. y /\ y e. {A})))
18	16, 17	syl5ib 223	. . 3 |- (Tr A -> ((x e. y /\ (y e. A \/ y e. {A})) -> (x e. A \/ x e. {A})))
19	18	19.21aivv 1665	. 2 |- (Tr A -> A.xA.y((x e. y /\ (y e. A \/ y e. {A})) -> (x e. A \/ x e. {A})))
20		df-suc 3663	. . . 4 |- suc A = (A u. {A})
21		treq 3417	. . . 4 |- (suc A = (A u. {A}) -> (Tr suc A <-> Tr (A u. {A})))
22	20, 21	ax-mp 7	. . 3 |- (Tr suc A <-> Tr (A u. {A}))
23		dftr2 3413	. . . 4 |- (Tr (A u. {A}) <-> A.xA.y((x e. y /\ y e. (A u. {A})) -> x e. (A u. {A})))
24		elun 2741	. . . . . . . 8 |- (y e. (A u. {A}) <-> (y e. A \/ y e. {A}))
25	24	anbi2i 538	. . . . . . 7 |- ((x e. y /\ y e. (A u. {A})) <-> (x e. y /\ (y e. A \/ y e. {A})))
26		elun 2741	. . . . . . 7 |- (x e. (A u. {A}) <-> (x e. A \/ x e. {A}))
27	25, 26	imbi12i 205	. . . . . 6 |- (((x e. y /\ y e. (A u. {A})) -> x e. (A u. {A})) <-> ((x e. y /\ (y e. A \/ y e. {A})) -> (x e. A \/ x e. {A})))
28	27	albii 1346	. . . . 5 |- (A.y((x e. y /\ y e. (A u. {A})) -> x e. (A u. {A})) <-> A.y((x e. y /\ (y e. A \/ y e. {A})) -> (x e. A \/ x e. {A})))
29	28	albii 1346	. . . 4 |- (A.xA.y((x e. y /\ y e. (A u. {A})) -> x e. (A u. {A})) <-> A.xA.y((x e. y /\ (y e. A \/ y e. {A})) -> (x e. A \/ x e. {A})))
30	23, 29	bitri 190	. . 3 |- (Tr (A u. {A}) <-> A.xA.y((x e. y /\ (y e. A \/ y e. {A})) -> (x e. A \/ x e. {A})))
31	22, 30	bitri 190	. 2 |- (Tr suc A <-> A.xA.y((x e. y /\ (y e. A \/ y e. {A})) -> (x e. A \/ x e. {A})))
32	19, 31	sylibr 217	1 |- (Tr A -> Tr suc A)

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300   u. cun 2591  {csn 3044  Tr wtr 3411  suc csuc 3659

This theorem is referenced by:  dfon2lem3 13851  dfon2lem7 13855

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

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-rex 2110  df-v 2294  df-un 2600  df-in 2603  df-ss 2605  df-sn 3049  df-uni 3178  df-tr 3412  df-suc 3663

Copyright terms: Public domain