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Theorem aomclem1 27151
Description: Lemma for dfac11 27160. 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 5539 . . 3  |-  ( R1
`  U. dom  z )  e.  _V
2 vex 2791 . . . . . . . 8  |-  z  e. 
_V
32dmex 4941 . . . . . . 7  |-  dom  z  e.  _V
43uniex 4516 . . . . . 6  |-  U. dom  z  e.  _V
54sucid 4471 . . . . 5  |-  U. dom  z  e.  suc  U. dom  z
6 aomclem1.su . . . . 5  |-  ( ph  ->  dom  z  =  suc  U.
dom  z )
75, 6syl5eleqr 2370 . . . 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 5525 . . . . . 6  |-  ( a  =  U. dom  z  ->  ( z `  a
)  =  ( z `
 U. dom  z
) )
10 fveq2 5525 . . . . . 6  |-  ( a  =  U. dom  z  ->  ( R1 `  a
)  =  ( R1
`  U. dom  z ) )
119, 10weeq12d 27136 . . . . 5  |-  ( a  =  U. dom  z  ->  ( ( z `  a )  We  ( R1 `  a )  <->  ( z `  U. dom  z )  We  ( R1 `  U. dom  z ) ) )
1211rspcva 2882 . . . 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 642 . . 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 27139 . . 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 644 . 2  |-  ( ph  ->  B  Or  ~P ( R1 `  U. dom  z
) )
176fveq2d 5529 . . . 4  |-  ( ph  ->  ( R1 `  dom  z )  =  ( R1 `  suc  U. dom  z ) )
18 aomclem1.on . . . . 5  |-  ( ph  ->  dom  z  e.  On )
19 onuni 4584 . . . . 5  |-  ( dom  z  e.  On  ->  U.
dom  z  e.  On )
20 r1suc 7442 . . . . 5  |-  ( U. dom  z  e.  On  ->  ( R1 `  suc  U.
dom  z )  =  ~P ( R1 `  U. dom  z ) )
2118, 19, 203syl 18 . . . 4  |-  ( ph  ->  ( R1 `  suc  U.
dom  z )  =  ~P ( R1 `  U. dom  z ) )
2217, 21eqtrd 2315 . . 3  |-  ( ph  ->  ( R1 `  dom  z )  =  ~P ( R1 `  U. dom  z ) )
23 soeq2 4334 . . 3  |-  ( ( R1 `  dom  z
)  =  ~P ( R1 `  U. dom  z
)  ->  ( B  Or  ( R1 `  dom  z )  <->  B  Or  ~P ( R1 `  U. dom  z ) ) )
2422, 23syl 15 . 2  |-  ( ph  ->  ( B  Or  ( R1 `  dom  z )  <-> 
B  Or  ~P ( R1 `  U. dom  z
) ) )
2516, 24mpbird 223 1  |-  ( ph  ->  B  Or  ( R1
`  dom  z )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023   {copab 4076    Or wor 4313    We wwe 4351   Oncon0 4392   suc csuc 4394   dom cdm 4689   ` cfv 5255   R1cr1 7434
This theorem is referenced by:  aomclem2  27152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-map 6774  df-r1 7436
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