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Theorem dfackm 8002
 Description: Equivalence of the Axiom of Choice and Maes' AC ackm 8301. The proof consists of lemmas kmlem1 7986 through kmlem16 8001 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 7965 with biid 228) establishes the AC equivalence shown by Maes' writeup. The left-hand-side AC shown here was chosen because it is shorter to display. (Contributed by NM, 13-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
dfackm CHOICE
Distinct variable group:   ,,,,

Proof of Theorem dfackm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac5 7965 . 2 CHOICE
2 eqid 2404 . . . . 5
32kmlem13 7998 . . . 4
4 kmlem8 7993 . . . . 5
54albii 1572 . . . 4
63, 5bitri 241 . . 3
7 df-ne 2569 . . . . . . . . 9
87bicomi 194 . . . . . . . 8
98anbi2i 676 . . . . . . 7
109anbi1i 677 . . . . . 6
1110imbi2i 304 . . . . 5
12 biid 228 . . . . 5
13 biid 228 . . . . 5
1411, 12, 13kmlem16 8001 . . . 4
1514albii 1572 . . 3
166, 15bitri 241 . 2
171, 16bitri 241 1 CHOICE
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359  wal 1546  wex 1547   wceq 1649   wcel 1721  weu 2254  cab 2390   wne 2567  wral 2666  wrex 2667   cdif 3277   cin 3279  c0 3588  csn 3774  cuni 3975  CHOICEwac 7952 This theorem is referenced by:  axac3  8300  ackm  8301  axac2  8302 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ac 7953
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