Theorem eltskg 9572

Description: Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.)

Assertion

Ref	Expression
eltskg	|- ( T e. V -> ( T e. Tarski <-> ( A. z e. T ( ~P z C_ T /\ E. w e. T ~P z C_ w ) /\ A. z e. ~P T ( z ~~ T \/ z e. T ) ) ) )

Distinct variable group:   w, T, z

Allowed substitution hints:   V( z, w)

Proof of Theorem eltskg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3627	. . . . 5 |- ( y = T -> ( ~P z C_ y <-> ~P z C_ T ) )
2		rexeq 3139	. . . . 5 |- ( y = T -> ( E. w e. y ~P z C_ w <-> E. w e. T ~P z C_ w ) )
3	1, 2	anbi12d 747	. . . 4 |- ( y = T -> ( ( ~P z C_ y /\ E. w e. y ~P z C_ w ) <-> ( ~P z C_ T /\ E. w e. T ~P z C_ w ) ) )
4	3	raleqbi1dv 3146	. . 3 |- ( y = T -> ( A. z e. y ( ~P z C_ y /\ E. w e. y ~P z C_ w ) <-> A. z e. T ( ~P z C_ T /\ E. w e. T ~P z C_ w ) ) )
5		pweq 4161	. . . 4 |- ( y = T -> ~P y = ~P T )
6		breq2 4657	. . . . 5 |- ( y = T -> ( z ~~ y <-> z ~~ T ) )
7		eleq2 2690	. . . . 5 |- ( y = T -> ( z e. y <-> z e. T ) )
8	6, 7	orbi12d 746	. . . 4 |- ( y = T -> ( ( z ~~ y \/ z e. y ) <-> ( z ~~ T \/ z e. T ) ) )
9	5, 8	raleqbidv 3152	. . 3 |- ( y = T -> ( A. z e. ~P y ( z ~~ y \/ z e. y ) <-> A. z e. ~P T ( z ~~ T \/ z e. T ) ) )
10	4, 9	anbi12d 747	. 2 |- ( y = T -> ( ( A. z e. y ( ~P z C_ y /\ E. w e. y ~P z C_ w ) /\ A. z e. ~P y ( z ~~ y \/ z e. y ) ) <-> ( A. z e. T ( ~P z C_ T /\ E. w e. T ~P z C_ w ) /\ A. z e. ~P T ( z ~~ T \/ z e. T ) ) ) )
11		df-tsk 9571	. 2 |- Tarski = { y | ( A. z e. y ( ~P z C_ y /\ E. w e. y ~P z C_ w ) /\ A. z e. ~P y ( z ~~ y \/ z e. y ) ) }
12	10, 11	elab2g 3353	1 |- ( T e. V -> ( T e. Tarski <-> ( A. z e. T ( ~P z C_ T /\ E. w e. T ~P z C_ w ) /\ A. z e. ~P T ( z ~~ T \/ z e. T ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   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

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:  eltsk2g  9573  tskpwss  9574  tsken  9576  grothtsk  9657