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Theorem aomclem1 27131
 Description: Lemma for dfac11 27139. This is the beginning of the proof that multiple choice is equivalent to choice. Our goal is to construct, by transfinite recursion, a well-ordering of . In what follows, is the index of the rank we wish to well-order, is the collection of well-orderings constructed so far, is the set of ordinal indexes of constructed ranks i.e. the next rank to construct, and is a postulated multiple-choice function. Successor case 1, define a simple ordering from the well-ordered predecessor. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypotheses
Ref Expression
aomclem1.b
aomclem1.on
aomclem1.su
aomclem1.we
Assertion
Ref Expression
aomclem1
Distinct variable group:   ,,,,
Allowed substitution hints:   (,,,,)   (,,,,)

Proof of Theorem aomclem1
StepHypRef Expression
1 fvex 5744 . . 3
2 vex 2961 . . . . . . . 8
32dmex 5134 . . . . . . 7
43uniex 4707 . . . . . 6
54sucid 4662 . . . . 5
6 aomclem1.su . . . . 5
75, 6syl5eleqr 2525 . . . 4
8 aomclem1.we . . . 4
9 fveq2 5730 . . . . . 6
10 fveq2 5730 . . . . . 6
119, 10weeq12d 27116 . . . . 5
1211rspcva 3052 . . . 4
137, 8, 12syl2anc 644 . . 3
14 aomclem1.b . . . 4
1514wepwso 27119 . . 3
161, 13, 15sylancr 646 . 2
176fveq2d 5734 . . . 4
18 aomclem1.on . . . . 5
19 onuni 4775 . . . . 5
20 r1suc 7698 . . . . 5
2118, 19, 203syl 19 . . . 4
2217, 21eqtrd 2470 . . 3
23 soeq2 4525 . . 3
2422, 23syl 16 . 2
2516, 24mpbird 225 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707  wrex 2708  cvv 2958  cpw 3801  cuni 4017   class class class wbr 4214  copab 4267   wor 4504   wwe 4542  con0 4583   csuc 4585   cdm 4880  cfv 5456  cr1 7690 This theorem is referenced by:  aomclem2  27132 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-map 7022  df-r1 7692
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