Theorem tartrel 15239

Description: A transitive element of a Tarski's class is a part of the class. JFM CLASSES2 th. 8

Assertion

Ref	Expression
tartrel	|- ((T e. Tarski /\ Tr A /\ A e. T) -> A C_ T)

Proof of Theorem tartrel

Step	Hyp	Ref	Expression
1		dftr4 3416	. . . 4 |- (Tr A <-> A C_ ~PA)
2		sstr2 2623	. . . . . . 7 |- (A C_ ~PA -> (~PA C_ T -> A C_ T))
3		tarax1 15216	. . . . . . 7 |- ((T e. Tarski /\ A e. T) -> ~PA C_ T)
4	2, 3	syl5com 63	. . . . . 6 |- ((T e. Tarski /\ A e. T) -> (A C_ ~PA -> A C_ T))
5	4	ex 402	. . . . 5 |- (T e. Tarski -> (A e. T -> (A C_ ~PA -> A C_ T)))
6	5	com3r 39	. . . 4 |- (A C_ ~PA -> (T e. Tarski -> (A e. T -> A C_ T)))
7	1, 6	sylbi 216	. . 3 |- (Tr A -> (T e. Tarski -> (A e. T -> A C_ T)))
8	7	com12 14	. 2 |- (T e. Tarski -> (Tr A -> (A e. T -> A C_ T)))
9	8	3imp 1061	1 |- ((T e. Tarski /\ Tr A /\ A e. T) -> A C_ T)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   e. wcel 1300   C_ wss 2593  ~Pcpw 3032  Tr wtr 3411   Tarski ctarski 15208

This theorem is referenced by:  tartord 15240  tarsuc3 15246  eltintpar 15276  inttaror 15277

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-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-uni 3178  df-br 3339  df-tr 3412  df-tsk 15210

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