Theorem tskss 9040

Description: The subsets of an element of a Tarski class belong to the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.)

Assertion

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

Proof of Theorem tskss

Step	Hyp	Ref	Expression
1		elpw2g 4566	. . . 4 |- ( A e. T -> ( B e. ~P A <-> B C_ A ) )
2	1	adantl 466	. . 3 |- ( ( T e. Tarski /\ A e. T ) -> ( B e. ~P A <-> B C_ A ) )
3		tskpwss 9034	. . . 4 |- ( ( T e. Tarski /\ A e. T ) -> ~P A C_ T )
4	3	sseld 3466	. . 3 |- ( ( T e. Tarski /\ A e. T ) -> ( B e. ~P A -> B e. T ) )
5	2, 4	sylbird 235	. 2 |- ( ( T e. Tarski /\ A e. T ) -> ( B C_ A -> B e. T ) )
6	5	3impia 1185	1 |- ( ( T e. Tarski /\ A e. T /\ B C_ A ) -> B e. T )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   e. wcel 1758   C_ wss 3439   ~Pcpw 3971   Tarskictsk 9030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-tsk 9031

This theorem is referenced by:  tskin  9041  tsksn  9042  tsksuc  9044  tsk0  9045  tskr1om2  9050  tskint  9067