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

Theorem ac10ct 8418
Description: A proof of the Well ordering theorem weth 8878, 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 3098 . . . . . 6  |-  y  e. 
_V
21brdom 7530 . . . . 5  |-  ( A  ~<_  y  <->  E. f  f : A -1-1-> y )
3 f1f 5771 . . . . . . . . . . . 12  |-  ( f : A -1-1-> y  -> 
f : A --> y )
4 frn 5727 . . . . . . . . . . . 12  |-  ( f : A --> y  ->  ran  f  C_  y )
53, 4syl 16 . . . . . . . . . . 11  |-  ( f : A -1-1-> y  ->  ran  f  C_  y )
6 onss 6611 . . . . . . . . . . 11  |-  ( y  e.  On  ->  y  C_  On )
7 sstr2 3496 . . . . . . . . . . 11  |-  ( ran  f  C_  y  ->  ( y  C_  On  ->  ran  f  C_  On )
)
85, 6, 7syl2im 38 . . . . . . . . . 10  |-  ( f : A -1-1-> y  -> 
( y  e.  On  ->  ran  f  C_  On ) )
9 epweon 6604 . . . . . . . . . 10  |-  _E  We  On
10 wess 4856 . . . . . . . . . 10  |-  ( ran  f  C_  On  ->  (  _E  We  On  ->  _E  We  ran  f ) )
118, 9, 10syl6mpi 62 . . . . . . . . 9  |-  ( f : A -1-1-> y  -> 
( y  e.  On  ->  _E  We  ran  f
) )
1211adantl 466 . . . . . . . 8  |-  ( ( A  ~<_  y  /\  f : A -1-1-> y )  -> 
( y  e.  On  ->  _E  We  ran  f
) )
13 f1f1orn 5817 . . . . . . . . . 10  |-  ( f : A -1-1-> y  -> 
f : A -1-1-onto-> ran  f
)
14 eqid 2443 . . . . . . . . . . 11  |-  { <. w ,  z >.  |  ( f `  w )  _E  ( f `  z ) }  =  { <. w ,  z
>.  |  ( f `  w )  _E  (
f `  z ) }
1514f1owe 6234 . . . . . . . . . 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 5057 . . . . . . . . . 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 7524 . . . . . . . . . . . 12  |-  Rel  ~<_
1918brrelexi 5030 . . . . . . . . . . 11  |-  ( A  ~<_  y  ->  A  e.  _V )
20 sqxpexg 6590 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  X.  A )  e. 
_V )
21 incom 3676 . . . . . . . . . . . 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
) )
22 inex1g 4580 . . . . . . . . . . . 12  |-  ( ( A  X.  A )  e.  _V  ->  (
( A  X.  A
)  i^i  { <. w ,  z >.  |  ( f `  w )  _E  ( f `  z ) } )  e.  _V )
2321, 22syl5eqelr 2536 . . . . . . . . . . 11  |-  ( ( A  X.  A )  e.  _V  ->  ( { <. w ,  z
>.  |  ( f `  w )  _E  (
f `  z ) }  i^i  ( A  X.  A ) )  e. 
_V )
24 weeq1 4857 . . . . . . . . . . . 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 ) )
2524spcegv 3181 . . . . . . . . . . 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 )
)
2619, 20, 23, 254syl 21 . . . . . . . . . 10  |-  ( A  ~<_  y  ->  ( ( { <. w ,  z
>.  |  ( f `  w )  _E  (
f `  z ) }  i^i  ( A  X.  A ) )  We  A  ->  E. x  x  We  A )
)
2717, 26syl5bi 217 . . . . . . . . 9  |-  ( A  ~<_  y  ->  ( { <. w ,  z >.  |  ( f `  w )  _E  (
f `  z ) }  We  A  ->  E. x  x  We  A
) )
2816, 27sylan9r 658 . . . . . . . 8  |-  ( ( A  ~<_  y  /\  f : A -1-1-> y )  -> 
(  _E  We  ran  f  ->  E. x  x  We  A ) )
2912, 28syld 44 . . . . . . 7  |-  ( ( A  ~<_  y  /\  f : A -1-1-> y )  -> 
( y  e.  On  ->  E. x  x  We  A ) )
3029impancom 440 . . . . . 6  |-  ( ( A  ~<_  y  /\  y  e.  On )  ->  (
f : A -1-1-> y  ->  E. x  x  We  A ) )
3130exlimdv 1711 . . . . 5  |-  ( ( A  ~<_  y  /\  y  e.  On )  ->  ( E. f  f : A -1-1-> y  ->  E. x  x  We  A )
)
322, 31syl5bi 217 . . . 4  |-  ( ( A  ~<_  y  /\  y  e.  On )  ->  ( A  ~<_  y  ->  E. x  x  We  A )
)
3332ex 434 . . 3  |-  ( A  ~<_  y  ->  ( y  e.  On  ->  ( A  ~<_  y  ->  E. x  x  We  A ) ) )
3433pm2.43b 50 . 2  |-  ( y  e.  On  ->  ( A  ~<_  y  ->  E. x  x  We  A )
)
3534rexlimiv 2929 1  |-  ( E. y  e.  On  A  ~<_  y  ->  E. x  x  We  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1599    e. wcel 1804   E.wrex 2794   _Vcvv 3095    i^i cin 3460    C_ wss 3461   class class class wbr 4437   {copab 4494    _E cep 4779    We wwe 4827   Oncon0 4868    X. cxp 4987   ran crn 4990   -->wf 5574   -1-1->wf1 5575   -1-1-onto->wf1o 5577   ` cfv 5578    ~<_ cdom 7516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-dom 7520
This theorem is referenced by:  ondomen  8421
  Copyright terms: Public domain W3C validator