Theorem grothtsk 9657

Description: The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)

Assertion

Ref	Expression
grothtsk	|- U. Tarski = _V

Proof of Theorem grothtsk

Dummy variables x w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axgroth5 9646	. . . . 5 |- E. x ( w e. x /\ A. y e. x ( ~P y C_ x /\ E. z e. x ~P y C_ z ) /\ A. y e. ~P x ( y ~~ x \/ y e. x ) )
2		vex 3203	. . . . . . . . 9 |- x e. _V
3		eltskg 9572	. . . . . . . . 9 |- ( x e. _V -> ( x e. Tarski <-> ( A. y e. x ( ~P y C_ x /\ E. z e. x ~P y C_ z ) /\ A. y e. ~P x ( y ~~ x \/ y e. x ) ) ) )
4	2, 3	ax-mp 5	. . . . . . . 8 |- ( x e. Tarski <-> ( A. y e. x ( ~P y C_ x /\ E. z e. x ~P y C_ z ) /\ A. y e. ~P x ( y ~~ x \/ y e. x ) ) )
5	4	anbi2i 730	. . . . . . 7 |- ( ( w e. x /\ x e. Tarski ) <-> ( w e. x /\ ( A. y e. x ( ~P y C_ x /\ E. z e. x ~P y C_ z ) /\ A. y e. ~P x ( y ~~ x \/ y e. x ) ) ) )
6		3anass 1042	. . . . . . 7 |- ( ( w e. x /\ A. y e. x ( ~P y C_ x /\ E. z e. x ~P y C_ z ) /\ A. y e. ~P x ( y ~~ x \/ y e. x ) ) <-> ( w e. x /\ ( A. y e. x ( ~P y C_ x /\ E. z e. x ~P y C_ z ) /\ A. y e. ~P x ( y ~~ x \/ y e. x ) ) ) )
7	5, 6	bitr4i 267	. . . . . 6 |- ( ( w e. x /\ x e. Tarski ) <-> ( w e. x /\ A. y e. x ( ~P y C_ x /\ E. z e. x ~P y C_ z ) /\ A. y e. ~P x ( y ~~ x \/ y e. x ) ) )
8	7	exbii 1774	. . . . 5 |- ( E. x ( w e. x /\ x e. Tarski ) <-> E. x ( w e. x /\ A. y e. x ( ~P y C_ x /\ E. z e. x ~P y C_ z ) /\ A. y e. ~P x ( y ~~ x \/ y e. x ) ) )
9	1, 8	mpbir 221	. . . 4 |- E. x ( w e. x /\ x e. Tarski )
10		eluni 4439	. . . 4 |- ( w e. U. Tarski <-> E. x ( w e. x /\ x e. Tarski ) )
11	9, 10	mpbir 221	. . 3 |- w e. U. Tarski
12		vex 3203	. . 3 |- w e. _V
13	11, 12	2th 254	. 2 |- ( w e. U. Tarski <-> w e. _V )
14	13	eqriv 2619	1 |- U. Tarski = _V

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   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-groth 9645

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-tsk 9571

This theorem is referenced by:  inaprc  9658  tskmval  9661  tskmcl  9663