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

Theorem noinfep 7574
Description: Using the Axiom of Regularity in the form zfregfr 7530, 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 7558 . . . . 5  |-  om  e.  _V
21mptex 5929 . . . 4  |-  ( w  e.  om  |->  ( F `
 w ) )  e.  _V
32rnex 5096 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  e.  _V
4 zfregfr 7530 . . 3  |-  _E  Fr  ran  ( w  e.  om  |->  ( F `  w ) )
5 ssid 3331 . . 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 5705 . . . . . . 7  |-  ( F `
 w )  e. 
_V
87a1i 11 . . . . . 6  |-  ( w  e.  om  ->  ( F `  w )  e.  _V )
96, 8mprg 2739 . . . . 5  |-  dom  (
w  e.  om  |->  ( F `  w ) )  =  om
10 peano1 4827 . . . . . 6  |-  (/)  e.  om
11 ne0i 3598 . . . . . 6  |-  ( (/)  e.  om  ->  om  =/=  (/) )
1210, 11ax-mp 8 . . . . 5  |-  om  =/=  (/)
139, 12eqnetri 2588 . . . 4  |-  dom  (
w  e.  om  |->  ( F `  w ) )  =/=  (/)
14 dm0rn0 5049 . . . . 5  |-  ( dom  ( w  e.  om  |->  ( F `  w ) )  =  (/)  <->  ran  ( w  e.  om  |->  ( F `
 w ) )  =  (/) )
1514necon3bii 2603 . . . 4  |-  ( dom  ( w  e.  om  |->  ( F `  w ) )  =/=  (/)  <->  ran  ( w  e.  om  |->  ( F `
 w ) )  =/=  (/) )
1613, 15mpbi 200 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  =/=  (/)
17 fri 4508 . . 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 655 . 2  |-  E. y  e.  ran  ( w  e. 
om  |->  ( F `  w ) ) A. z  e.  ran  ( w  e.  om  |->  ( F `
 w ) )  -.  z  _E  y
19 eqid 2408 . . . . . . 7  |-  ( w  e.  om  |->  ( F `
 w ) )  =  ( w  e. 
om  |->  ( F `  w ) )
207, 19fnmpti 5536 . . . . . 6  |-  ( w  e.  om  |->  ( F `
 w ) )  Fn  om
21 fvelrnb 5737 . . . . . 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 8 . . . . 5  |-  ( y  e.  ran  ( w  e.  om  |->  ( F `
 w ) )  <->  E. x  e.  om  ( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  y )
23 peano2 4828 . . . . . . . . . . 11  |-  ( x  e.  om  ->  suc  x  e.  om )
24 fveq2 5691 . . . . . . . . . . . 12  |-  ( w  =  suc  x  -> 
( F `  w
)  =  ( F `
 suc  x )
)
25 fvex 5705 . . . . . . . . . . . 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 5830 . . . . . . . . . . 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 645 . . . . . . . . . 10  |-  ( x  e.  om  ->  (
( w  e.  om  |->  ( F `  w ) ) `  suc  x
)  e.  ran  (
w  e.  om  |->  ( F `  w ) ) )
3027, 29eqeltrrd 2483 . . . . . . . . 9  |-  ( x  e.  om  ->  ( F `  suc  x )  e.  ran  ( w  e.  om  |->  ( F `
 w ) ) )
31 epel 4461 . . . . . . . . . . . . 13  |-  ( z  _E  y  <->  z  e.  y )
32 eleq1 2468 . . . . . . . . . . . . 13  |-  ( z  =  ( F `  suc  x )  ->  (
z  e.  y  <->  ( F `  suc  x )  e.  y ) )
3331, 32syl5bb 249 . . . . . . . . . . . 12  |-  ( z  =  ( F `  suc  x )  ->  (
z  _E  y  <->  ( F `  suc  x )  e.  y ) )
3433notbid 286 . . . . . . . . . . 11  |-  ( z  =  ( F `  suc  x )  ->  ( -.  z  _E  y  <->  -.  ( F `  suc  x )  e.  y ) )
35 df-nel 2574 . . . . . . . . . . 11  |-  ( ( F `  suc  x
)  e/  y  <->  -.  ( F `  suc  x )  e.  y )
3634, 35syl6bbr 255 . . . . . . . . . 10  |-  ( z  =  ( F `  suc  x )  ->  ( -.  z  _E  y  <->  ( F `  suc  x
)  e/  y )
)
3736rspccv 3013 . . . . . . . . 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 28 . . . . . . . 8  |-  ( x  e.  om  ->  ( A. z  e.  ran  ( w  e.  om  |->  ( F `  w ) )  -.  z  _E  y  ->  ( F `  suc  x )  e/  y ) )
39 fveq2 5691 . . . . . . . . . . . 12  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
40 fvex 5705 . . . . . . . . . . . 12  |-  ( F `
 x )  e. 
_V
4139, 19, 40fvmpt 5769 . . . . . . . . . . 11  |-  ( x  e.  om  ->  (
( w  e.  om  |->  ( F `  w ) ) `  x )  =  ( F `  x ) )
42 eqeq1 2414 . . . . . . . . . . 11  |-  ( ( ( w  e.  om  |->  ( F `  w ) ) `  x )  =  y  ->  (
( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  ( F `  x )  <-> 
y  =  ( F `
 x ) ) )
4341, 42syl5ibcom 212 . . . . . . . . . 10  |-  ( x  e.  om  ->  (
( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  y  ->  y  =  ( F `  x ) ) )
44 neleq2 2665 . . . . . . . . . . 11  |-  ( y  =  ( F `  x )  ->  (
( F `  suc  x )  e/  y  <->  ( F `  suc  x
)  e/  ( F `  x ) ) )
4544biimpd 199 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
( F `  suc  x )  e/  y  ->  ( F `  suc  x )  e/  ( F `  x )
) )
4643, 45syl6 31 . . . . . . . . 9  |-  ( x  e.  om  ->  (
( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  y  ->  ( ( F `
 suc  x )  e/  y  ->  ( F `
 suc  x )  e/  ( F `  x
) ) ) )
4746com23 74 . . . . . . . 8  |-  ( x  e.  om  ->  (
( F `  suc  x )  e/  y  ->  ( ( ( w  e.  om  |->  ( F `
 w ) ) `
 x )  =  y  ->  ( F `  suc  x )  e/  ( F `  x ) ) ) )
4838, 47syld 42 . . . . . . 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 29 . . . . . 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 2780 . . . . 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 209 . . . 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 29 . . 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 2788 . 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 8 1  |-  E. x  e.  om  ( F `  suc  x )  e/  ( F `  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721    =/= wne 2571    e/ wnel 2572   A.wral 2670   E.wrex 2671   _Vcvv 2920    C_ wss 3284   (/)c0 3592   class class class wbr 4176    e. cmpt 4230    _E cep 4456    Fr wfr 4502   suc csuc 4547   omcom 4808   dom cdm 4841   ran crn 4842    Fn wfn 5412   ` cfv 5417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664  ax-reg 7520  ax-inf2 7556
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425
  Copyright terms: Public domain W3C validator