Theorem elincin 15220

Description: The parts of an element of a Tarski's class are Tarski's classes.

Assertion

Ref	Expression
elincin	|- ((T e. Tarski /\ A e. T /\ B C_ A) -> B e. T)

Proof of Theorem elincin

Step	Hyp	Ref	Expression
1		ssexg 3457	. . . 4 |- ((B C_ A /\ A e. T) -> B e. _V)
2	1	ancoms 484	. . 3 |- ((A e. T /\ B C_ A) -> B e. _V)
3	2	3adant1 894	. 2 |- ((T e. Tarski /\ A e. T /\ B C_ A) -> B e. _V)
4		elpwg 3038	. . . . . . 7 |- (B e. _V -> (B e. ~PA <-> B C_ A))
5	4	biimprcd 173	. . . . . 6 |- (B C_ A -> (B e. _V -> B e. ~PA))
6	5	3ad2ant3 899	. . . . 5 |- ((T e. Tarski /\ A e. T /\ B C_ A) -> (B e. _V -> B e. ~PA))
7		tarax1 15216	. . . . . . 7 |- ((T e. Tarski /\ A e. T) -> ~PA C_ T)
8		ssel2 2616	. . . . . . . . 9 |- ((~PA C_ T /\ B e. ~PA) -> B e. T)
9	8	a1d 15	. . . . . . . 8 |- ((~PA C_ T /\ B e. ~PA) -> (B e. _V -> B e. T))
10	9	ex 402	. . . . . . 7 |- (~PA C_ T -> (B e. ~PA -> (B e. _V -> B e. T)))
11	7, 10	syl 12	. . . . . 6 |- ((T e. Tarski /\ A e. T) -> (B e. ~PA -> (B e. _V -> B e. T)))
12	11	3adant3 896	. . . . 5 |- ((T e. Tarski /\ A e. T /\ B C_ A) -> (B e. ~PA -> (B e. _V -> B e. T)))
13	6, 12	syld 30	. . . 4 |- ((T e. Tarski /\ A e. T /\ B C_ A) -> (B e. _V -> (B e. _V -> B e. T)))
14	13	com13 37	. . 3 |- (B e. _V -> (B e. _V -> ((T e. Tarski /\ A e. T /\ B C_ A) -> B e. T)))
15	14	pm2.43i 78	. 2 |- (B e. _V -> ((T e. Tarski /\ A e. T /\ B C_ A) -> B e. T))
16	3, 15	mpcom 60	1 |- ((T e. Tarski /\ A e. T /\ B C_ A) -> B e. T)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   e. wcel 1300  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032   Tarski ctarski 15208

This theorem is referenced by:  sexptrt 15243

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-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-un 2600  df-in 2603  df-ss 2605  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-tsk 15210

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