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Theorem ac10ct 7915
Description: A proof of the Well ordering theorem weth 8375, an Axiom of Choice equivalent, restricted to sets dominated by some ordinal (in particular finite sets and countable sets), proven in ZF without AC. (Contributed by Mario Carneiro, 5-Jan-2013.)
Assertion
Ref Expression
ac10ct  |-  ( E. y  e.  On  A  ~<_  y  ->  E. x  x  We  A )
Distinct variable group:    x, A, y

Proof of Theorem ac10ct
Dummy variables  f  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2959 . . . . . 6  |-  y  e. 
_V
21brdom 7120 . . . . 5  |-  ( A  ~<_  y  <->  E. f  f : A -1-1-> y )
3 f1f 5639 . . . . . . . . . . . 12  |-  ( f : A -1-1-> y  -> 
f : A --> y )
4 frn 5597 . . . . . . . . . . . 12  |-  ( f : A --> y  ->  ran  f  C_  y )
53, 4syl 16 . . . . . . . . . . 11  |-  ( f : A -1-1-> y  ->  ran  f  C_  y )
6 onss 4771 . . . . . . . . . . 11  |-  ( y  e.  On  ->  y  C_  On )
7 sstr2 3355 . . . . . . . . . . 11  |-  ( ran  f  C_  y  ->  ( y  C_  On  ->  ran  f  C_  On )
)
85, 6, 7syl2im 36 . . . . . . . . . 10  |-  ( f : A -1-1-> y  -> 
( y  e.  On  ->  ran  f  C_  On ) )
9 epweon 4764 . . . . . . . . . 10  |-  _E  We  On
10 wess 4569 . . . . . . . . . 10  |-  ( ran  f  C_  On  ->  (  _E  We  On  ->  _E  We  ran  f ) )
118, 9, 10syl6mpi 60 . . . . . . . . 9  |-  ( f : A -1-1-> y  -> 
( y  e.  On  ->  _E  We  ran  f
) )
1211adantl 453 . . . . . . . 8  |-  ( ( A  ~<_  y  /\  f : A -1-1-> y )  -> 
( y  e.  On  ->  _E  We  ran  f
) )
13 f1f1orn 5685 . . . . . . . . . 10  |-  ( f : A -1-1-> y  -> 
f : A -1-1-onto-> ran  f
)
14 eqid 2436 . . . . . . . . . . 11  |-  { <. w ,  z >.  |  ( f `  w )  _E  ( f `  z ) }  =  { <. w ,  z
>.  |  ( f `  w )  _E  (
f `  z ) }
1514f1owe 6073 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> ran  f  ->  (  _E  We  ran  f  ->  { <. w ,  z
>.  |  ( f `  w )  _E  (
f `  z ) }  We  A )
)
1613, 15syl 16 . . . . . . . . 9  |-  ( f : A -1-1-> y  -> 
(  _E  We  ran  f  ->  { <. w ,  z >.  |  ( f `  w )  _E  ( f `  z ) }  We  A ) )
17 weinxp 4945 . . . . . . . . . 10  |-  ( {
<. w ,  z >.  |  ( f `  w )  _E  (
f `  z ) }  We  A  <->  ( { <. w ,  z >.  |  ( f `  w )  _E  (
f `  z ) }  i^i  ( A  X.  A ) )  We  A )
18 reldom 7115 . . . . . . . . . . . . 13  |-  Rel  ~<_
1918brrelexi 4918 . . . . . . . . . . . 12  |-  ( A  ~<_  y  ->  A  e.  _V )
20 xpexg 4989 . . . . . . . . . . . . 13  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
2120anidms 627 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  ( A  X.  A )  e. 
_V )
2219, 21syl 16 . . . . . . . . . . 11  |-  ( A  ~<_  y  ->  ( A  X.  A )  e.  _V )
23 incom 3533 . . . . . . . . . . . 12  |-  ( ( A  X.  A )  i^i  { <. w ,  z >.  |  ( f `  w )  _E  ( f `  z ) } )  =  ( { <. w ,  z >.  |  ( f `  w )  _E  ( f `  z ) }  i^i  ( A  X.  A
) )
24 inex1g 4346 . . . . . . . . . . . 12  |-  ( ( A  X.  A )  e.  _V  ->  (
( A  X.  A
)  i^i  { <. w ,  z >.  |  ( f `  w )  _E  ( f `  z ) } )  e.  _V )
2523, 24syl5eqelr 2521 . . . . . . . . . . 11  |-  ( ( A  X.  A )  e.  _V  ->  ( { <. w ,  z
>.  |  ( f `  w )  _E  (
f `  z ) }  i^i  ( A  X.  A ) )  e. 
_V )
26 weeq1 4570 . . . . . . . . . . . 12  |-  ( x  =  ( { <. w ,  z >.  |  ( f `  w )  _E  ( f `  z ) }  i^i  ( A  X.  A
) )  ->  (
x  We  A  <->  ( { <. w ,  z >.  |  ( f `  w )  _E  (
f `  z ) }  i^i  ( A  X.  A ) )  We  A ) )
2726spcegv 3037 . . . . . . . . . . 11  |-  ( ( { <. w ,  z
>.  |  ( f `  w )  _E  (
f `  z ) }  i^i  ( A  X.  A ) )  e. 
_V  ->  ( ( {
<. w ,  z >.  |  ( f `  w )  _E  (
f `  z ) }  i^i  ( A  X.  A ) )  We  A  ->  E. x  x  We  A )
)
2822, 25, 273syl 19 . . . . . . . . . 10  |-  ( A  ~<_  y  ->  ( ( { <. w ,  z
>.  |  ( f `  w )  _E  (
f `  z ) }  i^i  ( A  X.  A ) )  We  A  ->  E. x  x  We  A )
)
2917, 28syl5bi 209 . . . . . . . . 9  |-  ( A  ~<_  y  ->  ( { <. w ,  z >.  |  ( f `  w )  _E  (
f `  z ) }  We  A  ->  E. x  x  We  A
) )
3016, 29sylan9r 640 . . . . . . . 8  |-  ( ( A  ~<_  y  /\  f : A -1-1-> y )  -> 
(  _E  We  ran  f  ->  E. x  x  We  A ) )
3112, 30syld 42 . . . . . . 7  |-  ( ( A  ~<_  y  /\  f : A -1-1-> y )  -> 
( y  e.  On  ->  E. x  x  We  A ) )
3231impancom 428 . . . . . 6  |-  ( ( A  ~<_  y  /\  y  e.  On )  ->  (
f : A -1-1-> y  ->  E. x  x  We  A ) )
3332exlimdv 1646 . . . . 5  |-  ( ( A  ~<_  y  /\  y  e.  On )  ->  ( E. f  f : A -1-1-> y  ->  E. x  x  We  A )
)
342, 33syl5bi 209 . . . 4  |-  ( ( A  ~<_  y  /\  y  e.  On )  ->  ( A  ~<_  y  ->  E. x  x  We  A )
)
3534ex 424 . . 3  |-  ( A  ~<_  y  ->  ( y  e.  On  ->  ( A  ~<_  y  ->  E. x  x  We  A ) ) )
3635pm2.43b 48 . 2  |-  ( y  e.  On  ->  ( A  ~<_  y  ->  E. x  x  We  A )
)
3736rexlimiv 2824 1  |-  ( E. y  e.  On  A  ~<_  y  ->  E. x  x  We  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    e. wcel 1725   E.wrex 2706   _Vcvv 2956    i^i cin 3319    C_ wss 3320   class class class wbr 4212   {copab 4265    _E cep 4492    We wwe 4540   Oncon0 4581    X. cxp 4876   ran crn 4879   -->wf 5450   -1-1->wf1 5451   -1-1-onto->wf1o 5453   ` cfv 5454    ~<_ cdom 7107
This theorem is referenced by:  ondomen  7918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-dom 7111
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