Theorem tskss 9086

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 4556	. . . 4 |- ( A e. T -> ( B e. ~P A <-> B C_ A ) )
2	1	adantl 464	. . 3 |- ( ( T e. Tarski /\ A e. T ) -> ( B e. ~P A <-> B C_ A ) )
3		tskpwss 9080	. . . 4 |- ( ( T e. Tarski /\ A e. T ) -> ~P A C_ T )
4	3	sseld 3440	. . 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 1194	1 |- ( ( T e. Tarski /\ A e. T /\ B C_ A ) -> B e. T )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   e. wcel 1842   C_ wss 3413   ~Pcpw 3954   Tarskictsk 9076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-tsk 9077

This theorem is referenced by:  tskin  9087  tsksn  9088  tsksuc  9090  tsk0  9091  tskr1om2  9096  tskint  9113