Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aomclem1 Unicode version

Theorem aomclem1 27254
Description: Lemma for dfac11 27263. 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 5555 . . 3  |-  ( R1
`  U. dom  z )  e.  _V
2 vex 2804 . . . . . . . 8  |-  z  e. 
_V
32dmex 4957 . . . . . . 7  |-  dom  z  e.  _V
43uniex 4532 . . . . . 6  |-  U. dom  z  e.  _V
54sucid 4487 . . . . 5  |-  U. dom  z  e.  suc  U. dom  z
6 aomclem1.su . . . . 5  |-  ( ph  ->  dom  z  =  suc  U.
dom  z )
75, 6syl5eleqr 2383 . . . 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 5541 . . . . . 6  |-  ( a  =  U. dom  z  ->  ( z `  a
)  =  ( z `
 U. dom  z
) )
10 fveq2 5541 . . . . . 6  |-  ( a  =  U. dom  z  ->  ( R1 `  a
)  =  ( R1
`  U. dom  z ) )
119, 10weeq12d 27239 . . . . 5  |-  ( a  =  U. dom  z  ->  ( ( z `  a )  We  ( R1 `  a )  <->  ( z `  U. dom  z )  We  ( R1 `  U. dom  z ) ) )
1211rspcva 2895 . . . 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 27242 . . 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 5545 . . . 4  |-  ( ph  ->  ( R1 `  dom  z )  =  ( R1 `  suc  U. dom  z ) )
18 aomclem1.on . . . . 5  |-  ( ph  ->  dom  z  e.  On )
19 onuni 4600 . . . . 5  |-  ( dom  z  e.  On  ->  U.
dom  z  e.  On )
20 r1suc 7458 . . . . 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 2328 . . 3  |-  ( ph  ->  ( R1 `  dom  z )  =  ~P ( R1 `  U. dom  z ) )
23 soeq2 4350 . . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039   {copab 4092    Or wor 4329    We wwe 4367   Oncon0 4408   suc csuc 4410   dom cdm 4705   ` cfv 5271   R1cr1 7450
This theorem is referenced by:  aomclem2  27255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-map 6790  df-r1 7452
  Copyright terms: Public domain W3C validator