MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ac10ct Unicode version

Theorem ac10ct 7677
Description: A proof of the Well ordering theorem weth 8138, 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 2804 . . . . . 6  |-  y  e. 
_V
21brdom 6890 . . . . 5  |-  ( A  ~<_  y  <->  E. f  f : A -1-1-> y )
3 f1f 5453 . . . . . . . . . . . 12  |-  ( f : A -1-1-> y  -> 
f : A --> y )
4 frn 5411 . . . . . . . . . . . 12  |-  ( f : A --> y  ->  ran  f  C_  y )
53, 4syl 15 . . . . . . . . . . 11  |-  ( f : A -1-1-> y  ->  ran  f  C_  y )
6 onss 4598 . . . . . . . . . . 11  |-  ( y  e.  On  ->  y  C_  On )
7 sstr2 3199 . . . . . . . . . . 11  |-  ( ran  f  C_  y  ->  ( y  C_  On  ->  ran  f  C_  On )
)
85, 6, 7syl2im 34 . . . . . . . . . 10  |-  ( f : A -1-1-> y  -> 
( y  e.  On  ->  ran  f  C_  On ) )
9 epweon 4591 . . . . . . . . . 10  |-  _E  We  On
10 wess 4396 . . . . . . . . . 10  |-  ( ran  f  C_  On  ->  (  _E  We  On  ->  _E  We  ran  f ) )
118, 9, 10syl6mpi 58 . . . . . . . . 9  |-  ( f : A -1-1-> y  -> 
( y  e.  On  ->  _E  We  ran  f
) )
1211adantl 452 . . . . . . . 8  |-  ( ( A  ~<_  y  /\  f : A -1-1-> y )  -> 
( y  e.  On  ->  _E  We  ran  f
) )
13 f1f1orn 5499 . . . . . . . . . 10  |-  ( f : A -1-1-> y  -> 
f : A -1-1-onto-> ran  f
)
14 eqid 2296 . . . . . . . . . . 11  |-  { <. w ,  z >.  |  ( f `  w )  _E  ( f `  z ) }  =  { <. w ,  z
>.  |  ( f `  w )  _E  (
f `  z ) }
1514f1owe 5866 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> ran  f  ->  (  _E  We  ran  f  ->  { <. w ,  z
>.  |  ( f `  w )  _E  (
f `  z ) }  We  A )
)
1613, 15syl 15 . . . . . . . . 9  |-  ( f : A -1-1-> y  -> 
(  _E  We  ran  f  ->  { <. w ,  z >.  |  ( f `  w )  _E  ( f `  z ) }  We  A ) )
17 weinxp 4773 . . . . . . . . . 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 6885 . . . . . . . . . . . . 13  |-  Rel  ~<_
1918brrelexi 4745 . . . . . . . . . . . 12  |-  ( A  ~<_  y  ->  A  e.  _V )
20 xpexg 4816 . . . . . . . . . . . . 13  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
2120anidms 626 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  ( A  X.  A )  e. 
_V )
2219, 21syl 15 . . . . . . . . . . 11  |-  ( A  ~<_  y  ->  ( A  X.  A )  e.  _V )
23 incom 3374 . . . . . . . . . . . 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 4173 . . . . . . . . . . . 12  |-  ( ( A  X.  A )  e.  _V  ->  (
( A  X.  A
)  i^i  { <. w ,  z >.  |  ( f `  w )  _E  ( f `  z ) } )  e.  _V )
2523, 24syl5eqelr 2381 . . . . . . . . . . 11  |-  ( ( A  X.  A )  e.  _V  ->  ( { <. w ,  z
>.  |  ( f `  w )  _E  (
f `  z ) }  i^i  ( A  X.  A ) )  e. 
_V )
26 weeq1 4397 . . . . . . . . . . . 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 2882 . . . . . . . . . . 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 18 . . . . . . . . . 10  |-  ( A  ~<_  y  ->  ( ( { <. w ,  z
>.  |  ( f `  w )  _E  (
f `  z ) }  i^i  ( A  X.  A ) )  We  A  ->  E. x  x  We  A )
)
2917, 28syl5bi 208 . . . . . . . . 9  |-  ( A  ~<_  y  ->  ( { <. w ,  z >.  |  ( f `  w )  _E  (
f `  z ) }  We  A  ->  E. x  x  We  A
) )
3016, 29sylan9r 639 . . . . . . . 8  |-  ( ( A  ~<_  y  /\  f : A -1-1-> y )  -> 
(  _E  We  ran  f  ->  E. x  x  We  A ) )
3112, 30syld 40 . . . . . . 7  |-  ( ( A  ~<_  y  /\  f : A -1-1-> y )  -> 
( y  e.  On  ->  E. x  x  We  A ) )
3231impancom 427 . . . . . 6  |-  ( ( A  ~<_  y  /\  y  e.  On )  ->  (
f : A -1-1-> y  ->  E. x  x  We  A ) )
3332exlimdv 1626 . . . . 5  |-  ( ( A  ~<_  y  /\  y  e.  On )  ->  ( E. f  f : A -1-1-> y  ->  E. x  x  We  A )
)
342, 33syl5bi 208 . . . 4  |-  ( ( A  ~<_  y  /\  y  e.  On )  ->  ( A  ~<_  y  ->  E. x  x  We  A )
)
3534ex 423 . . 3  |-  ( A  ~<_  y  ->  ( y  e.  On  ->  ( A  ~<_  y  ->  E. x  x  We  A ) ) )
3635pm2.43b 46 . 2  |-  ( y  e.  On  ->  ( A  ~<_  y  ->  E. x  x  We  A )
)
3736rexlimiv 2674 1  |-  ( E. y  e.  On  A  ~<_  y  ->  E. x  x  We  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    e. wcel 1696   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   class class class wbr 4039   {copab 4092    _E cep 4319    We wwe 4367   Oncon0 4408    X. cxp 4703   ran crn 4706   -->wf 5267   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271    ~<_ cdom 6877
This theorem is referenced by:  ondomen  7680
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-rab 2565  df-v 2803  df-sbc 3005  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-br 4040  df-opab 4094  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-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-dom 6881
  Copyright terms: Public domain W3C validator