Theorem dfac13 8539

Description: The axiom of choice holds iff every set has choice sequences as long as itself. (Contributed by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
dfac13	|- (CHOICE <-> A. x x e. AC x )

Proof of Theorem dfac13

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

Step	Hyp	Ref	Expression
1		vex 3112	. . . 4 |- x e. _V
2		acacni 8537	. . . . 5 |- ( (CHOICE /\ x e. _V ) -> AC x = _V )
3	1, 2	mpan2 671	. . . 4 |- (CHOICE -> AC x = _V )
4	1, 3	syl5eleqr 2552	. . 3 |- (CHOICE -> x e. AC x )
5	4	alrimiv 1720	. 2 |- (CHOICE -> A. x x e. AC x )
6		vex 3112	. . . . . . . . 9 |- z e. _V
7	6	pwex 4639	. . . . . . . 8 |- ~P z e. _V
8		id 22	. . . . . . . . 9 |- ( x = ~P z -> x = ~P z )
9		acneq 8441	. . . . . . . . 9 |- ( x = ~P z -> AC x = AC ~P z )
10	8, 9	eleq12d 2539	. . . . . . . 8 |- ( x = ~P z -> ( x e. AC x <-> ~P z e. AC ~P z ) )
11	7, 10	spcv 3200	. . . . . . 7 |- ( A. x x e. AC x -> ~P z e. AC ~P z )
12		vex 3112	. . . . . . . 8 |- y e. _V
13	6	canth2 7689	. . . . . . . . . 10 |- z ~< ~P z
14		sdomdom 7562	. . . . . . . . . 10 |- ( z ~< ~P z -> z ~<_ ~P z )
15		acndom2 8452	. . . . . . . . . 10 |- ( z ~<_ ~P z -> ( ~P z e. AC ~P z -> z e. AC ~P z ) )
16	13, 14, 15	mp2b 10	. . . . . . . . 9 |- ( ~P z e. AC ~P z -> z e. AC ~P z )
17		acnnum 8450	. . . . . . . . 9 |- ( z e. AC ~P z <-> z e. dom card )
18	16, 17	sylib 196	. . . . . . . 8 |- ( ~P z e. AC ~P z -> z e. dom card )
19		numacn 8447	. . . . . . . 8 |- ( y e. _V -> ( z e. dom card -> z e. AC y ) )
20	12, 18, 19	mpsyl 63	. . . . . . 7 |- ( ~P z e. AC ~P z -> z e. AC y )
21	11, 20	syl 16	. . . . . 6 |- ( A. x x e. AC x -> z e. AC y )
22	6	a1i 11	. . . . . 6 |- ( A. x x e. AC x -> z e. _V )
23	21, 22	2thd 240	. . . . 5 |- ( A. x x e. AC x -> ( z e. AC y <-> z e. _V ) )
24	23	eqrdv 2454	. . . 4 |- ( A. x x e. AC x -> AC y = _V )
25	24	alrimiv 1720	. . 3 |- ( A. x x e. AC x -> A. yAC y = _V )
26		dfacacn 8538	. . 3 |- (CHOICE <-> A. yAC y = _V )
27	25, 26	sylibr 212	. 2 |- ( A. x x e. AC x -> CHOICE )
28	5, 27	impbii 188	1 |- (CHOICE <-> A. x x e. AC x )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1393   = wceq 1395   e. wcel 1819   _Vcvv 3109   ~Pcpw 4015   class class class wbr 4456   dom cdm 5008   ~<_ cdom 7533   ~< csdm 7534   cardccrd 8333  AC wacn 8336  CHOICEwac 8513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-1o 7148  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-acn 8340  df-ac 8514

This theorem is referenced by: (None)