Theorem trintss 3427

Description: If A is transitive and non-null, then |^|A is a subset of A. (Contributed by Scott Fenton, 3-Mar-2011.)

Assertion

Ref	Expression
trintss	|- ((A =/= (/) /\ Tr A) -> |^|A C_ A)

Proof of Theorem trintss

Step	Hyp	Ref	Expression
1		r19.2z 2958	. . . . 5 |- ((A =/= (/) /\ A.x e. A y e. x) -> E.x e. A y e. x)
2	1	ex 402	. . . 4 |- (A =/= (/) -> (A.x e. A y e. x -> E.x e. A y e. x))
3		trel 3418	. . . . . . 7 |- (Tr A -> ((y e. x /\ x e. A) -> y e. A))
4	3	ancomsd 485	. . . . . 6 |- (Tr A -> ((x e. A /\ y e. x) -> y e. A))
5	4	exp3a 405	. . . . 5 |- (Tr A -> (x e. A -> (y e. x -> y e. A)))
6	5	r19.23adv 2215	. . . 4 |- (Tr A -> (E.x e. A y e. x -> y e. A))
7	2, 6	sylan9 517	. . 3 |- ((A =/= (/) /\ Tr A) -> (A.x e. A y e. x -> y e. A))
8		visset 2295	. . . 4 |- y e. _V
9	8	elint2 3221	. . 3 |- (y e. |^|A <-> A.x e. A y e. x)
10	7, 9	syl5ib 223	. 2 |- ((A =/= (/) /\ Tr A) -> (y e. |^|A -> y e. A))
11	10	ssrdv 2622	1 |- ((A =/= (/) /\ Tr A) -> |^|A C_ A)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   C_ wss 2593  (/)c0 2875  |^|cint 3214  Tr wtr 3411

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-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-uni 3178  df-int 3215  df-tr 3412

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