Theorem inttrp 14437

Description: The intersection of a non-empty element of a transitive class is a part of the class.

Assertion

Ref	Expression
inttrp	|- ((Tr A /\ B e. A /\ B =/= (/)) -> |^|B C_ A)

Proof of Theorem inttrp

Step	Hyp	Ref	Expression
1		trss 3421	. . . . 5 |- (Tr A -> (B e. A -> B C_ A))
2	1	imp 377	. . . 4 |- ((Tr A /\ B e. A) -> B C_ A)
3		df-tr 3412	. . . . . . . 8 |- (Tr A <-> U.A C_ A)
4		intssuni 3240	. . . . . . . . . . . . 13 |- (B =/= (/) -> |^|B C_ U.B)
5		sstr2 2623	. . . . . . . . . . . . 13 |- (|^|B C_ U.B -> (U.B C_ A -> |^|B C_ A))
6	4, 5	syl 12	. . . . . . . . . . . 12 |- (B =/= (/) -> (U.B C_ A -> |^|B C_ A))
7	6	3ad2ant3 899	. . . . . . . . . . 11 |- ((Tr A /\ B e. A /\ B =/= (/)) -> (U.B C_ A -> |^|B C_ A))
8		sstr 2625	. . . . . . . . . . 11 |- ((U.B C_ U.A /\ U.A C_ A) -> U.B C_ A)
9	7, 8	syl5com 63	. . . . . . . . . 10 |- ((U.B C_ U.A /\ U.A C_ A) -> ((Tr A /\ B e. A /\ B =/= (/)) -> |^|B C_ A))
10	9	ex 402	. . . . . . . . 9 |- (U.B C_ U.A -> (U.A C_ A -> ((Tr A /\ B e. A /\ B =/= (/)) -> |^|B C_ A)))
11	10	com3l 38	. . . . . . . 8 |- (U.A C_ A -> ((Tr A /\ B e. A /\ B =/= (/)) -> (U.B C_ U.A -> |^|B C_ A)))
12	3, 11	sylbi 216	. . . . . . 7 |- (Tr A -> ((Tr A /\ B e. A /\ B =/= (/)) -> (U.B C_ U.A -> |^|B C_ A)))
13	12	3ad2ant1 897	. . . . . 6 |- ((Tr A /\ B e. A /\ B =/= (/)) -> ((Tr A /\ B e. A /\ B =/= (/)) -> (U.B C_ U.A -> |^|B C_ A)))
14	13	pm2.43i 78	. . . . 5 |- ((Tr A /\ B e. A /\ B =/= (/)) -> (U.B C_ U.A -> |^|B C_ A))
15		uniss 3199	. . . . 5 |- (B C_ A -> U.B C_ U.A)
16	14, 15	syl5com 63	. . . 4 |- (B C_ A -> ((Tr A /\ B e. A /\ B =/= (/)) -> |^|B C_ A))
17	2, 16	syl 12	. . 3 |- ((Tr A /\ B e. A) -> ((Tr A /\ B e. A /\ B =/= (/)) -> |^|B C_ A))
18	17	3adant3 896	. 2 |- ((Tr A /\ B e. A /\ B =/= (/)) -> ((Tr A /\ B e. A /\ B =/= (/)) -> |^|B C_ A))
19	18	pm2.43i 78	1 |- ((Tr A /\ B e. A /\ B =/= (/)) -> |^|B C_ A)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   e. wcel 1300   =/= wne 2017   C_ wss 2593  (/)c0 2875  U.cuni 3177  |^|cint 3214  Tr wtr 3411

This theorem is referenced by:  intrtael 15256

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-3an 860  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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