Theorem dfac13 8511

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 3109	. . . 4 |- x e. _V
2		acacni 8509	. . . . 5 |- ( (CHOICE /\ x e. _V ) -> AC x = _V )
3	1, 2	mpan2 671	. . . 4 |- (CHOICE -> AC x = _V )
4	1, 3	syl5eleqr 2555	. . 3 |- (CHOICE -> x e. AC x )
5	4	alrimiv 1690	. 2 |- (CHOICE -> A. x x e. AC x )
6		vex 3109	. . . . . . . . 9 |- z e. _V
7	6	pwex 4623	. . . . . . . 8 |- ~P z e. _V
8		id 22	. . . . . . . . 9 |- ( x = ~P z -> x = ~P z )
9		acneq 8413	. . . . . . . . 9 |- ( x = ~P z -> AC x = AC ~P z )
10	8, 9	eleq12d 2542	. . . . . . . 8 |- ( x = ~P z -> ( x e. AC x <-> ~P z e. AC ~P z ) )
11	7, 10	spcv 3197	. . . . . . 7 |- ( A. x x e. AC x -> ~P z e. AC ~P z )
12		vex 3109	. . . . . . . 8 |- y e. _V
13	6	canth2 7660	. . . . . . . . . 10 |- z ~< ~P z
14		sdomdom 7533	. . . . . . . . . 10 |- ( z ~< ~P z -> z ~<_ ~P z )
15		acndom2 8424	. . . . . . . . . 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 8422	. . . . . . . . 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 8419	. . . . . . . 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 2457	. . . 4 |- ( A. x x e. AC x -> AC y = _V )
25	24	alrimiv 1690	. . 3 |- ( A. x x e. AC x -> A. yAC y = _V )
26		dfacacn 8510	. . 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 1372   = wceq 1374   e. wcel 1762   _Vcvv 3106   ~Pcpw 4003   class class class wbr 4440   dom cdm 4992   ~<_ cdom 7504   ~< csdm 7505   cardccrd 8305  AC wacn 8308  CHOICEwac 8485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-1o 7120  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-acn 8312  df-ac 8486

This theorem is referenced by: (None)