Theorem tsktrss 9583

Description: A transitive element of a Tarski class is a part of the class. JFM CLASSES2 th. 8. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 20-Sep-2014.)

Assertion

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

Proof of Theorem tsktrss

Step	Hyp	Ref	Expression
1		simp2 1062	. . 3 |- ( ( T e. Tarski /\ Tr A /\ A e. T ) -> Tr A )
2		dftr4 4757	. . 3 |- ( Tr A <-> A C_ ~P A )
3	1, 2	sylib 208	. 2 |- ( ( T e. Tarski /\ Tr A /\ A e. T ) -> A C_ ~P A )
4		tskpwss 9574	. . 3 |- ( ( T e. Tarski /\ A e. T ) -> ~P A C_ T )
5	4	3adant2 1080	. 2 |- ( ( T e. Tarski /\ Tr A /\ A e. T ) -> ~P A C_ T )
6	3, 5	sstrd 3613	1 |- ( ( T e. Tarski /\ Tr A /\ A e. T ) -> A C_ T )

Syntax hints:   -> wi 4   /\ w3a 1037   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   Tr wtr 4752   Tarskictsk 9570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-tr 4753  df-tsk 9571

This theorem is referenced by: (None)