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Theorem noinfep 7857
Description: Using the Axiom of Regularity in the form zfregfr 7810, 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 7841 . . . . 5  |-  om  e.  _V
21mptex 5943 . . . 4  |-  ( w  e.  om  |->  ( F `
 w ) )  e.  _V
32rnex 6507 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  e.  _V
4 zfregfr 7810 . . 3  |-  _E  Fr  ran  ( w  e.  om  |->  ( F `  w ) )
5 ssid 3370 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  C_  ran  ( w  e.  om  |->  ( F `
 w ) )
6 dmmptg 5330 . . . . . 6  |-  ( A. w  e.  om  ( F `  w )  e.  _V  ->  dom  ( w  e.  om  |->  ( F `
 w ) )  =  om )
7 fvex 5696 . . . . . . 7  |-  ( F `
 w )  e. 
_V
87a1i 11 . . . . . 6  |-  ( w  e.  om  ->  ( F `  w )  e.  _V )
96, 8mprg 2780 . . . . 5  |-  dom  (
w  e.  om  |->  ( F `  w ) )  =  om
10 peano1 6490 . . . . . 6  |-  (/)  e.  om
11 ne0i 3638 . . . . . 6  |-  ( (/)  e.  om  ->  om  =/=  (/) )
1210, 11ax-mp 5 . . . . 5  |-  om  =/=  (/)
139, 12eqnetri 2620 . . . 4  |-  dom  (
w  e.  om  |->  ( F `  w ) )  =/=  (/)
14 dm0rn0 5051 . . . . 5  |-  ( dom  ( w  e.  om  |->  ( F `  w ) )  =  (/)  <->  ran  ( w  e.  om  |->  ( F `
 w ) )  =  (/) )
1514necon3bii 2635 . . . 4  |-  ( dom  ( w  e.  om  |->  ( F `  w ) )  =/=  (/)  <->  ran  ( w  e.  om  |->  ( F `
 w ) )  =/=  (/) )
1613, 15mpbi 208 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  =/=  (/)
17 fri 4677 . . 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 )
183, 4, 5, 16, 17mp4an 673 . 2  |-  E. y  e.  ran  ( w  e. 
om  |->  ( F `  w ) ) A. z  e.  ran  ( w  e.  om  |->  ( F `
 w ) )  -.  z  _E  y
19 eqid 2438 . . . . . . 7  |-  ( w  e.  om  |->  ( F `
 w ) )  =  ( w  e. 
om  |->  ( F `  w ) )
207, 19fnmpti 5534 . . . . . 6  |-  ( w  e.  om  |->  ( F `
 w ) )  Fn  om
21 fvelrnb 5734 . . . . . 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 ) )
2220, 21ax-mp 5 . . . . 5  |-  ( y  e.  ran  ( w  e.  om  |->  ( F `
 w ) )  <->  E. x  e.  om  ( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  y )
23 peano2 6491 . . . . . . . . . . 11  |-  ( x  e.  om  ->  suc  x  e.  om )
24 fveq2 5686 . . . . . . . . . . . 12  |-  ( w  =  suc  x  -> 
( F `  w
)  =  ( F `
 suc  x )
)
25 fvex 5696 . . . . . . . . . . . 12  |-  ( F `
 suc  x )  e.  _V
2624, 19, 25fvmpt 5769 . . . . . . . . . . 11  |-  ( suc  x  e.  om  ->  ( ( w  e.  om  |->  ( F `  w ) ) `  suc  x
)  =  ( F `
 suc  x )
)
2723, 26syl 16 . . . . . . . . . 10  |-  ( x  e.  om  ->  (
( w  e.  om  |->  ( F `  w ) ) `  suc  x
)  =  ( F `
 suc  x )
)
28 fnfvelrn 5835 . . . . . . . . . . 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 ) ) )
2920, 23, 28sylancr 663 . . . . . . . . . 10  |-  ( x  e.  om  ->  (
( w  e.  om  |->  ( F `  w ) ) `  suc  x
)  e.  ran  (
w  e.  om  |->  ( F `  w ) ) )
3027, 29eqeltrrd 2513 . . . . . . . . 9  |-  ( x  e.  om  ->  ( F `  suc  x )  e.  ran  ( w  e.  om  |->  ( F `
 w ) ) )
31 epel 4630 . . . . . . . . . . . . 13  |-  ( z  _E  y  <->  z  e.  y )
32 eleq1 2498 . . . . . . . . . . . . 13  |-  ( z  =  ( F `  suc  x )  ->  (
z  e.  y  <->  ( F `  suc  x )  e.  y ) )
3331, 32syl5bb 257 . . . . . . . . . . . 12  |-  ( z  =  ( F `  suc  x )  ->  (
z  _E  y  <->  ( F `  suc  x )  e.  y ) )
3433notbid 294 . . . . . . . . . . 11  |-  ( z  =  ( F `  suc  x )  ->  ( -.  z  _E  y  <->  -.  ( F `  suc  x )  e.  y ) )
35 df-nel 2604 . . . . . . . . . . 11  |-  ( ( F `  suc  x
)  e/  y  <->  -.  ( F `  suc  x )  e.  y )
3634, 35syl6bbr 263 . . . . . . . . . 10  |-  ( z  =  ( F `  suc  x )  ->  ( -.  z  _E  y  <->  ( F `  suc  x
)  e/  y )
)
3736rspccv 3065 . . . . . . . . 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 ) )
3830, 37syl5com 30 . . . . . . . 8  |-  ( x  e.  om  ->  ( A. z  e.  ran  ( w  e.  om  |->  ( F `  w ) )  -.  z  _E  y  ->  ( F `  suc  x )  e/  y ) )
39 fveq2 5686 . . . . . . . . . . . 12  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
40 fvex 5696 . . . . . . . . . . . 12  |-  ( F `
 x )  e. 
_V
4139, 19, 40fvmpt 5769 . . . . . . . . . . 11  |-  ( x  e.  om  ->  (
( w  e.  om  |->  ( F `  w ) ) `  x )  =  ( F `  x ) )
42 eqeq1 2444 . . . . . . . . . . 11  |-  ( ( ( w  e.  om  |->  ( F `  w ) ) `  x )  =  y  ->  (
( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  ( F `  x )  <-> 
y  =  ( F `
 x ) ) )
4341, 42syl5ibcom 220 . . . . . . . . . 10  |-  ( x  e.  om  ->  (
( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  y  ->  y  =  ( F `  x ) ) )
44 neleq2 2705 . . . . . . . . . . 11  |-  ( y  =  ( F `  x )  ->  (
( F `  suc  x )  e/  y  <->  ( F `  suc  x
)  e/  ( F `  x ) ) )
4544biimpd 207 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
( F `  suc  x )  e/  y  ->  ( F `  suc  x )  e/  ( F `  x )
) )
4643, 45syl6 33 . . . . . . . . 9  |-  ( x  e.  om  ->  (
( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  y  ->  ( ( F `
 suc  x )  e/  y  ->  ( F `
 suc  x )  e/  ( F `  x
) ) ) )
4746com23 78 . . . . . . . 8  |-  ( x  e.  om  ->  (
( F `  suc  x )  e/  y  ->  ( ( ( w  e.  om  |->  ( F `
 w ) ) `
 x )  =  y  ->  ( F `  suc  x )  e/  ( F `  x ) ) ) )
4838, 47syld 44 . . . . . . 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 ) ) ) )
4948com12 31 . . . . . 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 ) ) ) )
5049reximdvai 2821 . . . . 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 )
) )
5122, 50syl5bi 217 . . . 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 )
) )
5251com12 31 . . 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 ) ) )
5352rexlimiv 2830 . 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 ) )
5418, 53ax-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 184    = wceq 1369    e. wcel 1756    =/= wne 2601    e/ wnel 2602   A.wral 2710   E.wrex 2711   _Vcvv 2967    C_ wss 3323   (/)c0 3632   class class class wbr 4287    e. cmpt 4345    _E cep 4625    Fr wfr 4671   suc csuc 4716   dom cdm 4835   ran crn 4836    Fn wfn 5408   ` cfv 5413   omcom 6471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367  ax-reg 7799  ax-inf2 7839
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472
This theorem is referenced by: (None)
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