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Theorem noinfep 8117
Description: Using the Axiom of Regularity in the form zfregfr 8070, show that there are no infinite descending 
e.-chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.)
Assertion
Ref Expression
noinfep  |-  E. x  e.  om  ( F `  suc  x )  e/  ( F `  x )
Distinct variable group:    x, F

Proof of Theorem noinfep
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omex 8101 . . . . 5  |-  om  e.  _V
21mptex 6095 . . . 4  |-  ( w  e.  om  |->  ( F `
 w ) )  e.  _V
32rnex 6685 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  e.  _V
4 zfregfr 8070 . . 3  |-  _E  Fr  ran  ( w  e.  om  |->  ( F `  w ) )
5 ssid 3426 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  C_  ran  ( w  e.  om  |->  ( F `
 w ) )
6 dmmptg 5294 . . . . . 6  |-  ( A. w  e.  om  ( F `  w )  e.  _V  ->  dom  ( w  e.  om  |->  ( F `
 w ) )  =  om )
7 fvex 5835 . . . . . . 7  |-  ( F `
 w )  e. 
_V
87a1i 11 . . . . . 6  |-  ( w  e.  om  ->  ( F `  w )  e.  _V )
96, 8mprg 2728 . . . . 5  |-  dom  (
w  e.  om  |->  ( F `  w ) )  =  om
10 peano1 6670 . . . . . 6  |-  (/)  e.  om
1110ne0ii 3711 . . . . 5  |-  om  =/=  (/)
129, 11eqnetri 2671 . . . 4  |-  dom  (
w  e.  om  |->  ( F `  w ) )  =/=  (/)
13 dm0rn0 5013 . . . . 5  |-  ( dom  ( w  e.  om  |->  ( F `  w ) )  =  (/)  <->  ran  ( w  e.  om  |->  ( F `
 w ) )  =  (/) )
1413necon3bii 2653 . . . 4  |-  ( dom  ( w  e.  om  |->  ( F `  w ) )  =/=  (/)  <->  ran  ( w  e.  om  |->  ( F `
 w ) )  =/=  (/) )
1512, 14mpbi 211 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  =/=  (/)
16 fri 4758 . . 3  |-  ( ( ( ran  ( w  e.  om  |->  ( F `
 w ) )  e.  _V  /\  _E  Fr  ran  ( w  e. 
om  |->  ( F `  w ) ) )  /\  ( ran  (
w  e.  om  |->  ( F `  w ) )  C_  ran  ( w  e.  om  |->  ( F `
 w ) )  /\  ran  ( w  e.  om  |->  ( F `
 w ) )  =/=  (/) ) )  ->  E. y  e.  ran  ( w  e.  om  |->  ( F `  w ) ) A. z  e. 
ran  ( w  e. 
om  |->  ( F `  w ) )  -.  z  _E  y )
173, 4, 5, 15, 16mp4an 677 . 2  |-  E. y  e.  ran  ( w  e. 
om  |->  ( F `  w ) ) A. z  e.  ran  ( w  e.  om  |->  ( F `
 w ) )  -.  z  _E  y
18 eqid 2428 . . . . . . 7  |-  ( w  e.  om  |->  ( F `
 w ) )  =  ( w  e. 
om  |->  ( F `  w ) )
197, 18fnmpti 5667 . . . . . 6  |-  ( w  e.  om  |->  ( F `
 w ) )  Fn  om
20 fvelrnb 5872 . . . . . 6  |-  ( ( w  e.  om  |->  ( F `  w ) )  Fn  om  ->  ( y  e.  ran  (
w  e.  om  |->  ( F `  w ) )  <->  E. x  e.  om  ( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  y ) )
2119, 20ax-mp 5 . . . . 5  |-  ( y  e.  ran  ( w  e.  om  |->  ( F `
 w ) )  <->  E. x  e.  om  ( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  y )
22 peano2 6671 . . . . . . . . . . 11  |-  ( x  e.  om  ->  suc  x  e.  om )
23 fveq2 5825 . . . . . . . . . . . 12  |-  ( w  =  suc  x  -> 
( F `  w
)  =  ( F `
 suc  x )
)
24 fvex 5835 . . . . . . . . . . . 12  |-  ( F `
 suc  x )  e.  _V
2523, 18, 24fvmpt 5908 . . . . . . . . . . 11  |-  ( suc  x  e.  om  ->  ( ( w  e.  om  |->  ( F `  w ) ) `  suc  x
)  =  ( F `
 suc  x )
)
2622, 25syl 17 . . . . . . . . . 10  |-  ( x  e.  om  ->  (
( w  e.  om  |->  ( F `  w ) ) `  suc  x
)  =  ( F `
 suc  x )
)
27 fnfvelrn 5978 . . . . . . . . . . 11  |-  ( ( ( w  e.  om  |->  ( F `  w ) )  Fn  om  /\  suc  x  e.  om )  ->  ( ( w  e. 
om  |->  ( F `  w ) ) `  suc  x )  e.  ran  ( w  e.  om  |->  ( F `  w ) ) )
2819, 22, 27sylancr 667 . . . . . . . . . 10  |-  ( x  e.  om  ->  (
( w  e.  om  |->  ( F `  w ) ) `  suc  x
)  e.  ran  (
w  e.  om  |->  ( F `  w ) ) )
2926, 28eqeltrrd 2507 . . . . . . . . 9  |-  ( x  e.  om  ->  ( F `  suc  x )  e.  ran  ( w  e.  om  |->  ( F `
 w ) ) )
30 epel 4710 . . . . . . . . . . . . 13  |-  ( z  _E  y  <->  z  e.  y )
31 eleq1 2494 . . . . . . . . . . . . 13  |-  ( z  =  ( F `  suc  x )  ->  (
z  e.  y  <->  ( F `  suc  x )  e.  y ) )
3230, 31syl5bb 260 . . . . . . . . . . . 12  |-  ( z  =  ( F `  suc  x )  ->  (
z  _E  y  <->  ( F `  suc  x )  e.  y ) )
3332notbid 295 . . . . . . . . . . 11  |-  ( z  =  ( F `  suc  x )  ->  ( -.  z  _E  y  <->  -.  ( F `  suc  x )  e.  y ) )
34 df-nel 2602 . . . . . . . . . . 11  |-  ( ( F `  suc  x
)  e/  y  <->  -.  ( F `  suc  x )  e.  y )
3533, 34syl6bbr 266 . . . . . . . . . 10  |-  ( z  =  ( F `  suc  x )  ->  ( -.  z  _E  y  <->  ( F `  suc  x
)  e/  y )
)
3635rspccv 3122 . . . . . . . . 9  |-  ( A. z  e.  ran  ( w  e.  om  |->  ( F `
 w ) )  -.  z  _E  y  ->  ( ( F `  suc  x )  e.  ran  ( w  e.  om  |->  ( F `  w ) )  ->  ( F `  suc  x )  e/  y ) )
3729, 36syl5com 31 . . . . . . . 8  |-  ( x  e.  om  ->  ( A. z  e.  ran  ( w  e.  om  |->  ( F `  w ) )  -.  z  _E  y  ->  ( F `  suc  x )  e/  y ) )
38 fveq2 5825 . . . . . . . . . . . 12  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
39 fvex 5835 . . . . . . . . . . . 12  |-  ( F `
 x )  e. 
_V
4038, 18, 39fvmpt 5908 . . . . . . . . . . 11  |-  ( x  e.  om  ->  (
( w  e.  om  |->  ( F `  w ) ) `  x )  =  ( F `  x ) )
41 eqeq1 2432 . . . . . . . . . . 11  |-  ( ( ( w  e.  om  |->  ( F `  w ) ) `  x )  =  y  ->  (
( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  ( F `  x )  <-> 
y  =  ( F `
 x ) ) )
4240, 41syl5ibcom 223 . . . . . . . . . 10  |-  ( x  e.  om  ->  (
( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  y  ->  y  =  ( F `  x ) ) )
43 neleq2 2707 . . . . . . . . . . 11  |-  ( y  =  ( F `  x )  ->  (
( F `  suc  x )  e/  y  <->  ( F `  suc  x
)  e/  ( F `  x ) ) )
4443biimpd 210 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
( F `  suc  x )  e/  y  ->  ( F `  suc  x )  e/  ( F `  x )
) )
4542, 44syl6 34 . . . . . . . . 9  |-  ( x  e.  om  ->  (
( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  y  ->  ( ( F `
 suc  x )  e/  y  ->  ( F `
 suc  x )  e/  ( F `  x
) ) ) )
4645com23 81 . . . . . . . 8  |-  ( x  e.  om  ->  (
( F `  suc  x )  e/  y  ->  ( ( ( w  e.  om  |->  ( F `
 w ) ) `
 x )  =  y  ->  ( F `  suc  x )  e/  ( F `  x ) ) ) )
4737, 46syld 45 . . . . . . 7  |-  ( x  e.  om  ->  ( A. z  e.  ran  ( w  e.  om  |->  ( F `  w ) )  -.  z  _E  y  ->  ( (
( w  e.  om  |->  ( F `  w ) ) `  x )  =  y  ->  ( F `  suc  x )  e/  ( F `  x ) ) ) )
4847com12 32 . . . . . 6  |-  ( A. z  e.  ran  ( w  e.  om  |->  ( F `
 w ) )  -.  z  _E  y  ->  ( x  e.  om  ->  ( ( ( w  e.  om  |->  ( F `
 w ) ) `
 x )  =  y  ->  ( F `  suc  x )  e/  ( F `  x ) ) ) )
4948reximdvai 2836 . . . . 5  |-  ( A. z  e.  ran  ( w  e.  om  |->  ( F `
 w ) )  -.  z  _E  y  ->  ( E. x  e. 
om  ( ( w  e.  om  |->  ( F `
 w ) ) `
 x )  =  y  ->  E. x  e.  om  ( F `  suc  x )  e/  ( F `  x )
) )
5021, 49syl5bi 220 . . . 4  |-  ( A. z  e.  ran  ( w  e.  om  |->  ( F `
 w ) )  -.  z  _E  y  ->  ( y  e.  ran  ( w  e.  om  |->  ( F `  w ) )  ->  E. x  e.  om  ( F `  suc  x )  e/  ( F `  x )
) )
5150com12 32 . . 3  |-  ( y  e.  ran  ( w  e.  om  |->  ( F `
 w ) )  ->  ( A. z  e.  ran  ( w  e. 
om  |->  ( F `  w ) )  -.  z  _E  y  ->  E. x  e.  om  ( F `  suc  x
)  e/  ( F `  x ) ) )
5251rexlimiv 2850 . 2  |-  ( E. y  e.  ran  (
w  e.  om  |->  ( F `  w ) ) A. z  e. 
ran  ( w  e. 
om  |->  ( F `  w ) )  -.  z  _E  y  ->  E. x  e.  om  ( F `  suc  x
)  e/  ( F `  x ) )
5317, 52ax-mp 5 1  |-  E. x  e.  om  ( F `  suc  x )  e/  ( F `  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1872    =/= wne 2599    e/ wnel 2600   A.wral 2714   E.wrex 2715   _Vcvv 3022    C_ wss 3379   (/)c0 3704   class class class wbr 4366    |-> cmpt 4425    _E cep 4705    Fr wfr 4752   dom cdm 4796   ran crn 4797   suc csuc 5387    Fn wfn 5539   ` cfv 5544   omcom 6650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541  ax-reg 8060  ax-inf2 8099
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-om 6651
This theorem is referenced by: (None)
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