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Theorem dfac11 27128
 Description: The right-hand side of this theorem (compare with ac4 8347), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 7552, despite not mentioning the cumulative hierarchy in any way as most consequences of regularity do. This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it. A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.)
Assertion
Ref Expression
dfac11 CHOICE
Distinct variable group:   ,,

Proof of Theorem dfac11
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac3 7994 . . 3 CHOICE
2 raleq 2896 . . . . . 6
32exbidv 1636 . . . . 5
43cbvalv 1984 . . . 4
5 neeq1 2606 . . . . . . . . . 10
6 fveq2 5720 . . . . . . . . . . 11
7 id 20 . . . . . . . . . . 11
86, 7eleq12d 2503 . . . . . . . . . 10
95, 8imbi12d 312 . . . . . . . . 9
109cbvralv 2924 . . . . . . . 8
11 fveq2 5720 . . . . . . . . . . . . . . 15
1211sneqd 3819 . . . . . . . . . . . . . 14
13 eqid 2435 . . . . . . . . . . . . . 14
14 snex 4397 . . . . . . . . . . . . . 14
1512, 13, 14fvmpt 5798 . . . . . . . . . . . . 13
16153ad2ant1 978 . . . . . . . . . . . 12
17 simp3 959 . . . . . . . . . . . . . . . 16
1817snssd 3935 . . . . . . . . . . . . . . 15
1914elpw 3797 . . . . . . . . . . . . . . 15
2018, 19sylibr 204 . . . . . . . . . . . . . 14
21 snfi 7179 . . . . . . . . . . . . . . 15
2221a1i 11 . . . . . . . . . . . . . 14
23 elin 3522 . . . . . . . . . . . . . 14
2420, 22, 23sylanbrc 646 . . . . . . . . . . . . 13
25 fvex 5734 . . . . . . . . . . . . . . 15
2625snnz 3914 . . . . . . . . . . . . . 14
2726a1i 11 . . . . . . . . . . . . 13
28 eldifsn 3919 . . . . . . . . . . . . 13
2924, 27, 28sylanbrc 646 . . . . . . . . . . . 12
3016, 29eqeltrd 2509 . . . . . . . . . . 11
31303exp 1152 . . . . . . . . . 10
3231a2d 24 . . . . . . . . 9
3332ralimia 2771 . . . . . . . 8
3410, 33sylbi 188 . . . . . . 7
35 vex 2951 . . . . . . . . 9
3635mptex 5958 . . . . . . . 8
37 fveq1 5719 . . . . . . . . . . 11
3837eleq1d 2501 . . . . . . . . . 10
3938imbi2d 308 . . . . . . . . 9
4039ralbidv 2717 . . . . . . . 8
4136, 40spcev 3035 . . . . . . 7
4234, 41syl 16 . . . . . 6
4342exlimiv 1644 . . . . 5
4443alimi 1568 . . . 4
454, 44sylbi 188 . . 3
461, 45sylbi 188 . 2 CHOICE
47 fvex 5734 . . . . . . 7
4847pwex 4374 . . . . . 6
49 raleq 2896 . . . . . . 7
5049exbidv 1636 . . . . . 6
5148, 50spcv 3034 . . . . 5
52 rankon 7713 . . . . . . . 8
5352a1i 11 . . . . . . 7
54 id 20 . . . . . . 7
5553, 54aomclem8 27127 . . . . . 6
5655exlimiv 1644 . . . . 5
57 vex 2951 . . . . . 6
58 r1rankid 7777 . . . . . 6
59 wess 4561 . . . . . . 7
6059eximdv 1632 . . . . . 6
6157, 58, 60mp2b 10 . . . . 5
6251, 56, 613syl 19 . . . 4
6362alrimiv 1641 . . 3
64 dfac8 8007 . . 3 CHOICE
6563, 64sylibr 204 . 2 CHOICE
6646, 65impbii 181 1 CHOICE
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   w3a 936  wal 1549  wex 1550   wceq 1652   wcel 1725   wne 2598  wral 2697  cvv 2948   cdif 3309   cin 3311   wss 3312  c0 3620  cpw 3791  csn 3806   cmpt 4258   wwe 4532  con0 4573  cfv 5446  cfn 7101  cr1 7680  crnk 7681  CHOICEwac 7988 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-reg 7552  ax-inf2 7588 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-er 6897  df-map 7012  df-en 7102  df-fin 7105  df-sup 7438  df-r1 7682  df-rank 7683  df-card 7818  df-ac 7989
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