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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  ( R1 `  A ). In what follows,  A is the index of the rank we wish to well-order,  z is the collection of well-orderings constructed so far,  dom  z is the set of ordinal indexes of constructed ranks i.e. the next rank to construct, and  y 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  |-  B  =  { <. a ,  b
>.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a
)  /\  A. d  e.  ( R1 `  U. dom  z ) ( d ( z `  U. dom  z ) c  -> 
( d  e.  a  <-> 
d  e.  b ) ) ) }
aomclem1.on  |-  ( ph  ->  dom  z  e.  On )
aomclem1.su  |-  ( ph  ->  dom  z  =  suc  U.
dom  z )
aomclem1.we  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
Assertion
Ref Expression
aomclem1  |-  ( ph  ->  B  Or  ( R1
`  dom  z )
)
Distinct variable group:    z, a, b, c, d
Allowed substitution hints:    ph( z, a, b, c, d)    B( z, a, b, c, d)

Proof of Theorem aomclem1
StepHypRef Expression
1 fvex 5744 . . 3  |-  ( R1
`  U. dom  z )  e.  _V
2 vex 2961 . . . . . . . 8  |-  z  e. 
_V
32dmex 5134 . . . . . . 7  |-  dom  z  e.  _V
43uniex 4707 . . . . . 6  |-  U. dom  z  e.  _V
54sucid 4662 . . . . 5  |-  U. dom  z  e.  suc  U. dom  z
6 aomclem1.su . . . . 5  |-  ( ph  ->  dom  z  =  suc  U.
dom  z )
75, 6syl5eleqr 2525 . . . 4  |-  ( ph  ->  U. dom  z  e. 
dom  z )
8 aomclem1.we . . . 4  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
9 fveq2 5730 . . . . . 6  |-  ( a  =  U. dom  z  ->  ( z `  a
)  =  ( z `
 U. dom  z
) )
10 fveq2 5730 . . . . . 6  |-  ( a  =  U. dom  z  ->  ( R1 `  a
)  =  ( R1
`  U. dom  z ) )
119, 10weeq12d 27116 . . . . 5  |-  ( a  =  U. dom  z  ->  ( ( z `  a )  We  ( R1 `  a )  <->  ( z `  U. dom  z )  We  ( R1 `  U. dom  z ) ) )
1211rspcva 3052 . . . 4  |-  ( ( U. dom  z  e. 
dom  z  /\  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )  -> 
( z `  U. dom  z )  We  ( R1 `  U. dom  z
) )
137, 8, 12syl2anc 644 . . 3  |-  ( ph  ->  ( z `  U. dom  z )  We  ( R1 `  U. dom  z
) )
14 aomclem1.b . . . 4  |-  B  =  { <. a ,  b
>.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a
)  /\  A. d  e.  ( R1 `  U. dom  z ) ( d ( z `  U. dom  z ) c  -> 
( d  e.  a  <-> 
d  e.  b ) ) ) }
1514wepwso 27119 . . 3  |-  ( ( ( R1 `  U. dom  z )  e.  _V  /\  ( z `  U. dom  z )  We  ( R1 `  U. dom  z
) )  ->  B  Or  ~P ( R1 `  U. dom  z ) )
161, 13, 15sylancr 646 . 2  |-  ( ph  ->  B  Or  ~P ( R1 `  U. dom  z
) )
176fveq2d 5734 . . . 4  |-  ( ph  ->  ( R1 `  dom  z )  =  ( R1 `  suc  U. dom  z ) )
18 aomclem1.on . . . . 5  |-  ( ph  ->  dom  z  e.  On )
19 onuni 4775 . . . . 5  |-  ( dom  z  e.  On  ->  U.
dom  z  e.  On )
20 r1suc 7698 . . . . 5  |-  ( U. dom  z  e.  On  ->  ( R1 `  suc  U.
dom  z )  =  ~P ( R1 `  U. dom  z ) )
2118, 19, 203syl 19 . . . 4  |-  ( ph  ->  ( R1 `  suc  U.
dom  z )  =  ~P ( R1 `  U. dom  z ) )
2217, 21eqtrd 2470 . . 3  |-  ( ph  ->  ( R1 `  dom  z )  =  ~P ( R1 `  U. dom  z ) )
23 soeq2 4525 . . 3  |-  ( ( R1 `  dom  z
)  =  ~P ( R1 `  U. dom  z
)  ->  ( B  Or  ( R1 `  dom  z )  <->  B  Or  ~P ( R1 `  U. dom  z ) ) )
2422, 23syl 16 . 2  |-  ( ph  ->  ( B  Or  ( R1 `  dom  z )  <-> 
B  Or  ~P ( R1 `  U. dom  z
) ) )
2516, 24mpbird 225 1  |-  ( ph  ->  B  Or  ( R1
`  dom  z )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958   ~Pcpw 3801   U.cuni 4017   class class class wbr 4214   {copab 4267    Or wor 4504    We wwe 4542   Oncon0 4583   suc csuc 4585   dom cdm 4880   ` cfv 5456   R1cr1 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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