Theorem dfackm 8567

Description: Equivalence of the Axiom of Choice and Maes' AC ackm 8866. The proof consists of lemmas kmlem1 8551 through kmlem16 8566 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 8530 with biid 236) 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

Distinct variable group:   ,,,,

Proof of Theorem dfackm

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac5 8530	. 2
2		eqid 2457	. . . . 5
3	2	kmlem13 8563	. . . 4
4		kmlem8 8558	. . . . 5
5	4	albii 1640	. . . 4
6	3, 5	bitri 249	. . 3
7		df-ne 2654	. . . . . . . . 9
8	7	bicomi 202	. . . . . . . 8
9	8	anbi2i 694	. . . . . . 7
10	9	anbi1i 695	. . . . . 6
11	10	imbi2i 312	. . . . 5
12		biid 236	. . . . 5
13		biid 236	. . . . 5
14	11, 12, 13	kmlem16 8566	. . . 4
15	14	albii 1640	. . 3
16	6, 15	bitri 249	. 2
17	1, 16	bitri 249	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  E!weu 2282  {cab 2442  =/=wne 2652  A.wral 2807  E.wrex 2808  \cdif 3472  i^icin 3474  c0 3784  {csn 4029  U.cuni 4249  wac 8517

This theorem is referenced by:  axac3  8865  ackm  8866  axac2  8867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ac 8518