Theorem ttac 37603

Description: Tarski's theorem about choice: infxpidm 9384 is equivalent to ax-ac 9281. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)

Assertion

Ref	Expression
ttac	|- (CHOICE <-> A. c ( om ~<_ c -> ( c X. c ) ~~ c ) )

Proof of Theorem ttac

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac10 8959	. 2 |- (CHOICE <-> dom card = _V )
2		vex 3203	. . . . . 6 |- c e. _V
3		eleq2 2690	. . . . . 6 |- ( dom card = _V -> ( c e. dom card <-> c e. _V ) )
4	2, 3	mpbiri 248	. . . . 5 |- ( dom card = _V -> c e. dom card )
5		infxpidm2 8840	. . . . . 6 |- ( ( c e. dom card /\ om ~<_ c ) -> ( c X. c ) ~~ c )
6	5	ex 450	. . . . 5 |- ( c e. dom card -> ( om ~<_ c -> ( c X. c ) ~~ c ) )
7	4, 6	syl 17	. . . 4 |- ( dom card = _V -> ( om ~<_ c -> ( c X. c ) ~~ c ) )
8	7	alrimiv 1855	. . 3 |- ( dom card = _V -> A. c ( om ~<_ c -> ( c X. c ) ~~ c ) )
9		finnum 8774	. . . . . . 7 |- ( a e. Fin -> a e. dom card )
10	9	adantl 482	. . . . . 6 |- ( ( A. c ( om ~<_ c -> ( c X. c ) ~~ c ) /\ a e. Fin ) -> a e. dom card )
11		harcl 8466	. . . . . . . . 9 |- (har ` a ) e. On
12		onenon 8775	. . . . . . . . 9 |- ( (har ` a ) e. On -> (har ` a ) e. dom card )
13	11, 12	ax-mp 5	. . . . . . . 8 |- (har ` a ) e. dom card
14		fvex 6201	. . . . . . . . . . . . . 14 |- (har ` a ) e. _V
15		vex 3203	. . . . . . . . . . . . . 14 |- a e. _V
16	14, 15	unex 6956	. . . . . . . . . . . . 13 |- ( (har ` a ) u. a ) e. _V
17		harinf 37601	. . . . . . . . . . . . . . 15 |- ( ( a e. _V /\ -. a e. Fin ) -> om C_ (har ` a ) )
18	15, 17	mpan 706	. . . . . . . . . . . . . 14 |- ( -. a e. Fin -> om C_ (har ` a ) )
19		ssun1 3776	. . . . . . . . . . . . . 14 |- (har ` a ) C_ ( (har ` a ) u. a )
20	18, 19	syl6ss 3615	. . . . . . . . . . . . 13 |- ( -. a e. Fin -> om C_ ( (har ` a ) u. a ) )
21		ssdomg 8001	. . . . . . . . . . . . 13 |- ( ( (har ` a ) u. a ) e. _V -> ( om C_ ( (har ` a ) u. a ) -> om ~<_ ( (har ` a ) u. a ) ) )
22	16, 20, 21	mpsyl 68	. . . . . . . . . . . 12 |- ( -. a e. Fin -> om ~<_ ( (har ` a ) u. a ) )
23		breq2 4657	. . . . . . . . . . . . . 14 |- ( c = ( (har ` a ) u. a ) -> ( om ~<_ c <-> om ~<_ ( (har ` a ) u. a ) ) )
24		xpeq12 5134	. . . . . . . . . . . . . . . 16 |- ( ( c = ( (har ` a ) u. a ) /\ c = ( (har ` a ) u. a ) ) -> ( c X. c ) = ( ( (har ` a ) u. a ) X. ( (har ` a ) u. a ) ) )
25	24	anidms 677	. . . . . . . . . . . . . . 15 |- ( c = ( (har ` a ) u. a ) -> ( c X. c ) = ( ( (har ` a ) u. a ) X. ( (har ` a ) u. a ) ) )
26		id 22	. . . . . . . . . . . . . . 15 |- ( c = ( (har ` a ) u. a ) -> c = ( (har ` a ) u. a ) )
27	25, 26	breq12d 4666	. . . . . . . . . . . . . 14 |- ( c = ( (har ` a ) u. a ) -> ( ( c X. c ) ~~ c <-> ( ( (har ` a ) u. a ) X. ( (har ` a ) u. a ) ) ~~ ( (har ` a ) u. a ) ) )
28	23, 27	imbi12d 334	. . . . . . . . . . . . 13 |- ( c = ( (har ` a ) u. a ) -> ( ( om ~<_ c -> ( c X. c ) ~~ c ) <-> ( om ~<_ ( (har ` a ) u. a ) -> ( ( (har ` a ) u. a ) X. ( (har ` a ) u. a ) ) ~~ ( (har ` a ) u. a ) ) ) )
29	16, 28	spcv 3299	. . . . . . . . . . . 12 |- ( A. c ( om ~<_ c -> ( c X. c ) ~~ c ) -> ( om ~<_ ( (har ` a ) u. a ) -> ( ( (har ` a ) u. a ) X. ( (har ` a ) u. a ) ) ~~ ( (har ` a ) u. a ) ) )
30	22, 29	syl5 34	. . . . . . . . . . 11 |- ( A. c ( om ~<_ c -> ( c X. c ) ~~ c ) -> ( -. a e. Fin -> ( ( (har ` a ) u. a ) X. ( (har ` a ) u. a ) ) ~~ ( (har ` a ) u. a ) ) )
31	30	imp 445	. . . . . . . . . 10 |- ( ( A. c ( om ~<_ c -> ( c X. c ) ~~ c ) /\ -. a e. Fin ) -> ( ( (har ` a ) u. a ) X. ( (har ` a ) u. a ) ) ~~ ( (har ` a ) u. a ) )
32		harndom 8469	. . . . . . . . . . . 12 |- -. (har ` a ) ~<_ a
33		ssdomg 8001	. . . . . . . . . . . . . 14 |- ( ( (har ` a ) u. a ) e. _V -> ( (har ` a ) C_ ( (har ` a ) u. a ) -> (har ` a ) ~<_ ( (har ` a ) u. a ) ) )
34	16, 19, 33	mp2 9	. . . . . . . . . . . . 13 |- (har ` a ) ~<_ ( (har ` a ) u. a )
35		domtr 8009	. . . . . . . . . . . . 13 |- ( ( (har ` a ) ~<_ ( (har ` a ) u. a ) /\ ( (har ` a ) u. a ) ~<_ a ) -> (har ` a ) ~<_ a )
36	34, 35	mpan 706	. . . . . . . . . . . 12 |- ( ( (har ` a ) u. a ) ~<_ a -> (har ` a ) ~<_ a )
37	32, 36	mto 188	. . . . . . . . . . 11 |- -. ( (har ` a ) u. a ) ~<_ a
38		unxpwdom2 8493	. . . . . . . . . . 11 |- ( ( ( (har ` a ) u. a ) X. ( (har ` a ) u. a ) ) ~~ ( (har ` a ) u. a ) -> ( ( (har ` a ) u. a ) ~<_* (har ` a ) \/ ( (har ` a ) u. a ) ~<_ a ) )
39		orel2 398	. . . . . . . . . . 11 |- ( -. ( (har ` a ) u. a ) ~<_ a -> ( ( ( (har ` a ) u. a ) ~<_* (har ` a ) \/ ( (har ` a ) u. a ) ~<_ a ) -> ( (har ` a ) u. a ) ~<_* (har ` a ) ) )
40	37, 38, 39	mpsyl 68	. . . . . . . . . 10 |- ( ( ( (har ` a ) u. a ) X. ( (har ` a ) u. a ) ) ~~ ( (har ` a ) u. a ) -> ( (har ` a ) u. a ) ~<_* (har ` a ) )
41	31, 40	syl 17	. . . . . . . . 9 |- ( ( A. c ( om ~<_ c -> ( c X. c ) ~~ c ) /\ -. a e. Fin ) -> ( (har ` a ) u. a ) ~<_* (har ` a ) )
42		wdomnumr 8887	. . . . . . . . . 10 |- ( (har ` a ) e. dom card -> ( ( (har ` a ) u. a ) ~<_* (har ` a ) <-> ( (har ` a ) u. a ) ~<_ (har ` a ) ) )
43	13, 42	ax-mp 5	. . . . . . . . 9 |- ( ( (har ` a ) u. a ) ~<_* (har ` a ) <-> ( (har ` a ) u. a ) ~<_ (har ` a ) )
44	41, 43	sylib 208	. . . . . . . 8 |- ( ( A. c ( om ~<_ c -> ( c X. c ) ~~ c ) /\ -. a e. Fin ) -> ( (har ` a ) u. a ) ~<_ (har ` a ) )
45		numdom 8861	. . . . . . . 8 |- ( ( (har ` a ) e. dom card /\ ( (har ` a ) u. a ) ~<_ (har ` a ) ) -> ( (har ` a ) u. a ) e. dom card )
46	13, 44, 45	sylancr 695	. . . . . . 7 |- ( ( A. c ( om ~<_ c -> ( c X. c ) ~~ c ) /\ -. a e. Fin ) -> ( (har ` a ) u. a ) e. dom card )
47		ssun2 3777	. . . . . . 7 |- a C_ ( (har ` a ) u. a )
48		ssnum 8862	. . . . . . 7 |- ( ( ( (har ` a ) u. a ) e. dom card /\ a C_ ( (har ` a ) u. a ) ) -> a e. dom card )
49	46, 47, 48	sylancl 694	. . . . . 6 |- ( ( A. c ( om ~<_ c -> ( c X. c ) ~~ c ) /\ -. a e. Fin ) -> a e. dom card )
50	10, 49	pm2.61dan 832	. . . . 5 |- ( A. c ( om ~<_ c -> ( c X. c ) ~~ c ) -> a e. dom card )
51	50	alrimiv 1855	. . . 4 |- ( A. c ( om ~<_ c -> ( c X. c ) ~~ c ) -> A. a a e. dom card )
52		eqv 3205	. . . 4 |- ( dom card = _V <-> A. a a e. dom card )
53	51, 52	sylibr 224	. . 3 |- ( A. c ( om ~<_ c -> ( c X. c ) ~~ c ) -> dom card = _V )
54	8, 53	impbii 199	. 2 |- ( dom card = _V <-> A. c ( om ~<_ c -> ( c X. c ) ~~ c ) )
55	1, 54	bitri 264	1 |- (CHOICE <-> A. c ( om ~<_ c -> ( c X. c ) ~~ c ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Oncon0 5723   ` cfv 5888   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955  harchar 8461   ~<_^* cwdom 8462   cardccrd 8761  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  ax-inf2 8538

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-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-wdom 8464  df-card 8765  df-acn 8768  df-ac 8939

This theorem is referenced by: (None)