Theorem dfac13 8964

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 3203	. . . 4 |- x e. _V
2		acacni 8962	. . . . 5 |- ( (CHOICE /\ x e. _V ) -> AC x = _V )
3	1, 2	mpan2 707	. . . 4 |- (CHOICE -> AC x = _V )
4	1, 3	syl5eleqr 2708	. . 3 |- (CHOICE -> x e. AC x )
5	4	alrimiv 1855	. 2 |- (CHOICE -> A. x x e. AC x )
6		vpwex 4849	. . . . . . . 8 |- ~P z e. _V
7		id 22	. . . . . . . . 9 |- ( x = ~P z -> x = ~P z )
8		acneq 8866	. . . . . . . . 9 |- ( x = ~P z -> AC x = AC ~P z )
9	7, 8	eleq12d 2695	. . . . . . . 8 |- ( x = ~P z -> ( x e. AC x <-> ~P z e. AC ~P z ) )
10	6, 9	spcv 3299	. . . . . . 7 |- ( A. x x e. AC x -> ~P z e. AC ~P z )
11		vex 3203	. . . . . . . 8 |- y e. _V
12		vex 3203	. . . . . . . . . . 11 |- z e. _V
13	12	canth2 8113	. . . . . . . . . 10 |- z ~< ~P z
14		sdomdom 7983	. . . . . . . . . 10 |- ( z ~< ~P z -> z ~<_ ~P z )
15		acndom2 8877	. . . . . . . . . 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 8875	. . . . . . . . 9 |- ( z e. AC ~P z <-> z e. dom card )
18	16, 17	sylib 208	. . . . . . . 8 |- ( ~P z e. AC ~P z -> z e. dom card )
19		numacn 8872	. . . . . . . 8 |- ( y e. _V -> ( z e. dom card -> z e. AC y ) )
20	11, 18, 19	mpsyl 68	. . . . . . 7 |- ( ~P z e. AC ~P z -> z e. AC y )
21	10, 20	syl 17	. . . . . 6 |- ( A. x x e. AC x -> z e. AC y )
22	12	a1i 11	. . . . . 6 |- ( A. x x e. AC x -> z e. _V )
23	21, 22	2thd 255	. . . . 5 |- ( A. x x e. AC x -> ( z e. AC y <-> z e. _V ) )
24	23	eqrdv 2620	. . . 4 |- ( A. x x e. AC x -> AC y = _V )
25	24	alrimiv 1855	. . 3 |- ( A. x x e. AC x -> A. yAC y = _V )
26		dfacacn 8963	. . 3 |- (CHOICE <-> A. yAC y = _V )
27	25, 26	sylibr 224	. 2 |- ( A. x x e. AC x -> CHOICE )
28	5, 27	impbii 199	1 |- (CHOICE <-> A. x x e. AC x )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761  AC wacn 8764  CHOICEwac 8938

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

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-lim 5728  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-1o 7560  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-acn 8768  df-ac 8939

This theorem is referenced by: (None)