Theorem grothac 9652

Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 9291). This can be put in a more conventional form via ween 8858 and dfac8 8957. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
grothac	|- dom card = _V

Proof of Theorem grothac

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

Step	Hyp	Ref	Expression
1		pweq 4161	. . . . . . . . . 10 |- ( x = y -> ~P x = ~P y )
2	1	sseq1d 3632	. . . . . . . . 9 |- ( x = y -> ( ~P x C_ u <-> ~P y C_ u ) )
3	1	eleq1d 2686	. . . . . . . . 9 |- ( x = y -> ( ~P x e. u <-> ~P y e. u ) )
4	2, 3	anbi12d 747	. . . . . . . 8 |- ( x = y -> ( ( ~P x C_ u /\ ~P x e. u ) <-> ( ~P y C_ u /\ ~P y e. u ) ) )
5	4	rspcva 3307	. . . . . . 7 |- ( ( y e. u /\ A. x e. u ( ~P x C_ u /\ ~P x e. u ) ) -> ( ~P y C_ u /\ ~P y e. u ) )
6	5	simpld 475	. . . . . 6 |- ( ( y e. u /\ A. x e. u ( ~P x C_ u /\ ~P x e. u ) ) -> ~P y C_ u )
7		rabss 3679	. . . . . . 7 |- ( { x e. ~P u | x ~< u } C_ u <-> A. x e. ~P u ( x ~< u -> x e. u ) )
8	7	biimpri 218	. . . . . 6 |- ( A. x e. ~P u ( x ~< u -> x e. u ) -> { x e. ~P u | x ~< u } C_ u )
9		vex 3203	. . . . . . . . . 10 |- y e. _V
10	9	canth2 8113	. . . . . . . . 9 |- y ~< ~P y
11		sdomdom 7983	. . . . . . . . 9 |- ( y ~< ~P y -> y ~<_ ~P y )
12	10, 11	ax-mp 5	. . . . . . . 8 |- y ~<_ ~P y
13		vex 3203	. . . . . . . . 9 |- u e. _V
14		ssdomg 8001	. . . . . . . . 9 |- ( u e. _V -> ( ~P y C_ u -> ~P y ~<_ u ) )
15	13, 14	ax-mp 5	. . . . . . . 8 |- ( ~P y C_ u -> ~P y ~<_ u )
16		domtr 8009	. . . . . . . 8 |- ( ( y ~<_ ~P y /\ ~P y ~<_ u ) -> y ~<_ u )
17	12, 15, 16	sylancr 695	. . . . . . 7 |- ( ~P y C_ u -> y ~<_ u )
18		tskwe 8776	. . . . . . . 8 |- ( ( u e. _V /\ { x e. ~P u | x ~< u } C_ u ) -> u e. dom card )
19	13, 18	mpan 706	. . . . . . 7 |- ( { x e. ~P u | x ~< u } C_ u -> u e. dom card )
20		numdom 8861	. . . . . . . 8 |- ( ( u e. dom card /\ y ~<_ u ) -> y e. dom card )
21	20	expcom 451	. . . . . . 7 |- ( y ~<_ u -> ( u e. dom card -> y e. dom card ) )
22	17, 19, 21	syl2im 40	. . . . . 6 |- ( ~P y C_ u -> ( { x e. ~P u | x ~< u } C_ u -> y e. dom card ) )
23	6, 8, 22	syl2im 40	. . . . 5 |- ( ( y e. u /\ A. x e. u ( ~P x C_ u /\ ~P x e. u ) ) -> ( A. x e. ~P u ( x ~< u -> x e. u ) -> y e. dom card ) )
24	23	3impia 1261	. . . 4 |- ( ( y e. u /\ A. x e. u ( ~P x C_ u /\ ~P x e. u ) /\ A. x e. ~P u ( x ~< u -> x e. u ) ) -> y e. dom card )
25		axgroth6 9650	. . . 4 |- E. u ( y e. u /\ A. x e. u ( ~P x C_ u /\ ~P x e. u ) /\ A. x e. ~P u ( x ~< u -> x e. u ) )
26	24, 25	exlimiiv 1859	. . 3 |- y e. dom card
27	26, 9	2th 254	. 2 |- ( y e. dom card <-> y e. _V )
28	27	eqriv 2619	1 |- dom card = _V

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-groth 9645

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-wrecs 7407  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765

This theorem is referenced by:  axgroth3  9653