Theorem tskpw 9575

Description: Second axiom of a Tarski class. The powerset of an element of a Tarski class belongs to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
tskpw	|- ( ( T e. Tarski /\ A e. T ) -> ~P A e. T )

Proof of Theorem tskpw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eltsk2g 9573	. . . . 5 |- ( T e. Tarski -> ( T e. Tarski <-> ( A. x e. T ( ~P x C_ T /\ ~P x e. T ) /\ A. x e. ~P T ( x ~~ T \/ x e. T ) ) ) )
2	1	ibi 256	. . . 4 |- ( T e. Tarski -> ( A. x e. T ( ~P x C_ T /\ ~P x e. T ) /\ A. x e. ~P T ( x ~~ T \/ x e. T ) ) )
3	2	simpld 475	. . 3 |- ( T e. Tarski -> A. x e. T ( ~P x C_ T /\ ~P x e. T ) )
4		simpr 477	. . . 4 |- ( ( ~P x C_ T /\ ~P x e. T ) -> ~P x e. T )
5	4	ralimi 2952	. . 3 |- ( A. x e. T ( ~P x C_ T /\ ~P x e. T ) -> A. x e. T ~P x e. T )
6	3, 5	syl 17	. 2 |- ( T e. Tarski -> A. x e. T ~P x e. T )
7		pweq 4161	. . . 4 |- ( x = A -> ~P x = ~P A )
8	7	eleq1d 2686	. . 3 |- ( x = A -> ( ~P x e. T <-> ~P A e. T ) )
9	8	rspccva 3308	. 2 |- ( ( A. x e. T ~P x e. T /\ A e. T ) -> ~P A e. T )
10	6, 9	sylan 488	1 |- ( ( T e. Tarski /\ A e. T ) -> ~P A e. T )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   ~~ cen 7952   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  ax-sep 4781  ax-pow 4843

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-br 4654  df-tsk 9571

This theorem is referenced by:  tsksn  9582  tsksuc  9584  tskr1om  9589  inttsk  9596  tskcard  9603  tskwun  9606  grutsk1  9643  pwinfi3  37868