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

Theorem noinfep 7362
Description: Using the Axiom of Regularity in the form zfregfr 7318, 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 7346 . . . . 5  |-  om  e.  _V
21mptex 5748 . . . 4  |-  ( w  e.  om  |->  ( F `
 w ) )  e.  _V
32rnex 4944 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  e.  _V
4 zfregfr 7318 . . 3  |-  _E  Fr  ran  ( w  e.  om  |->  ( F `  w ) )
5 ssid 3199 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  C_  ran  ( w  e.  om  |->  ( F `
 w ) )
6 dmmptg 5172 . . . . . 6  |-  ( A. w  e.  om  ( F `  w )  e.  _V  ->  dom  ( w  e.  om  |->  ( F `
 w ) )  =  om )
7 fvex 5541 . . . . . . 7  |-  ( F `
 w )  e. 
_V
87a1i 10 . . . . . 6  |-  ( w  e.  om  ->  ( F `  w )  e.  _V )
96, 8mprg 2614 . . . . 5  |-  dom  (
w  e.  om  |->  ( F `  w ) )  =  om
10 peano1 4677 . . . . . 6  |-  (/)  e.  om
11 ne0i 3463 . . . . . 6  |-  ( (/)  e.  om  ->  om  =/=  (/) )
1210, 11ax-mp 8 . . . . 5  |-  om  =/=  (/)
139, 12eqnetri 2465 . . . 4  |-  dom  (
w  e.  om  |->  ( F `  w ) )  =/=  (/)
14 dm0rn0 4897 . . . . 5  |-  ( dom  ( w  e.  om  |->  ( F `  w ) )  =  (/)  <->  ran  ( w  e.  om  |->  ( F `
 w ) )  =  (/) )
1514necon3bii 2480 . . . 4  |-  ( dom  ( w  e.  om  |->  ( F `  w ) )  =/=  (/)  <->  ran  ( w  e.  om  |->  ( F `
 w ) )  =/=  (/) )
1613, 15mpbi 199 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  =/=  (/)
17 fri 4357 . . 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 654 . 2  |-  E. y  e.  ran  ( w  e. 
om  |->  ( F `  w ) ) A. z  e.  ran  ( w  e.  om  |->  ( F `
 w ) )  -.  z  _E  y
19 eqid 2285 . . . . . . 7  |-  ( w  e.  om  |->  ( F `
 w ) )  =  ( w  e. 
om  |->  ( F `  w ) )
207, 19fnmpti 5374 . . . . . 6  |-  ( w  e.  om  |->  ( F `
 w ) )  Fn  om
21 fvelrnb 5572 . . . . . 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 4678 . . . . . . . . . . 11  |-  ( x  e.  om  ->  suc  x  e.  om )
24 fveq2 5527 . . . . . . . . . . . 12  |-  ( w  =  suc  x  -> 
( F `  w
)  =  ( F `
 suc  x )
)
25 fvex 5541 . . . . . . . . . . . 12  |-  ( F `
 suc  x )  e.  _V
2624, 19, 25fvmpt 5604 . . . . . . . . . . 11  |-  ( suc  x  e.  om  ->  ( ( w  e.  om  |->  ( F `  w ) ) `  suc  x
)  =  ( F `
 suc  x )
)
2723, 26syl 15 . . . . . . . . . 10  |-  ( x  e.  om  ->  (
( w  e.  om  |->  ( F `  w ) ) `  suc  x
)  =  ( F `
 suc  x )
)
28 fnfvelrn 5664 . . . . . . . . . . 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 644 . . . . . . . . . 10  |-  ( x  e.  om  ->  (
( w  e.  om  |->  ( F `  w ) ) `  suc  x
)  e.  ran  (
w  e.  om  |->  ( F `  w ) ) )
3027, 29eqeltrrd 2360 . . . . . . . . 9  |-  ( x  e.  om  ->  ( F `  suc  x )  e.  ran  ( w  e.  om  |->  ( F `
 w ) ) )
31 epel 4310 . . . . . . . . . . . . 13  |-  ( z  _E  y  <->  z  e.  y )
32 eleq1 2345 . . . . . . . . . . . . 13  |-  ( z  =  ( F `  suc  x )  ->  (
z  e.  y  <->  ( F `  suc  x )  e.  y ) )
3331, 32syl5bb 248 . . . . . . . . . . . 12  |-  ( z  =  ( F `  suc  x )  ->  (
z  _E  y  <->  ( F `  suc  x )  e.  y ) )
3433notbid 285 . . . . . . . . . . 11  |-  ( z  =  ( F `  suc  x )  ->  ( -.  z  _E  y  <->  -.  ( F `  suc  x )  e.  y ) )
35 df-nel 2451 . . . . . . . . . . 11  |-  ( ( F `  suc  x
)  e/  y  <->  -.  ( F `  suc  x )  e.  y )
3634, 35syl6bbr 254 . . . . . . . . . 10  |-  ( z  =  ( F `  suc  x )  ->  ( -.  z  _E  y  <->  ( F `  suc  x
)  e/  y )
)
3736rspccv 2883 . . . . . . . . 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 26 . . . . . . . 8  |-  ( x  e.  om  ->  ( A. z  e.  ran  ( w  e.  om  |->  ( F `  w ) )  -.  z  _E  y  ->  ( F `  suc  x )  e/  y ) )
39 fveq2 5527 . . . . . . . . . . . 12  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
40 fvex 5541 . . . . . . . . . . . 12  |-  ( F `
 x )  e. 
_V
4139, 19, 40fvmpt 5604 . . . . . . . . . . 11  |-  ( x  e.  om  ->  (
( w  e.  om  |->  ( F `  w ) ) `  x )  =  ( F `  x ) )
42 eqeq1 2291 . . . . . . . . . . 11  |-  ( ( ( w  e.  om  |->  ( F `  w ) ) `  x )  =  y  ->  (
( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  ( F `  x )  <-> 
y  =  ( F `
 x ) ) )
4341, 42syl5ibcom 211 . . . . . . . . . 10  |-  ( x  e.  om  ->  (
( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  y  ->  y  =  ( F `  x ) ) )
44 neleq2 2540 . . . . . . . . . . 11  |-  ( y  =  ( F `  x )  ->  (
( F `  suc  x )  e/  y  <->  ( F `  suc  x
)  e/  ( F `  x ) ) )
4544biimpd 198 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
( F `  suc  x )  e/  y  ->  ( F `  suc  x )  e/  ( F `  x )
) )
4643, 45syl6 29 . . . . . . . . 9  |-  ( x  e.  om  ->  (
( ( w  e. 
om  |->  ( F `  w ) ) `  x )  =  y  ->  ( ( F `
 suc  x )  e/  y  ->  ( F `
 suc  x )  e/  ( F `  x
) ) ) )
4746com23 72 . . . . . . . 8  |-  ( x  e.  om  ->  (
( F `  suc  x )  e/  y  ->  ( ( ( w  e.  om  |->  ( F `
 w ) ) `
 x )  =  y  ->  ( F `  suc  x )  e/  ( F `  x ) ) ) )
4838, 47syld 40 . . . . . . 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 27 . . . . . 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 2655 . . . . 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 208 . . . 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 27 . . 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 2663 . 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 176    = wceq 1625    e. wcel 1686    =/= wne 2448    e/ wnel 2449   A.wral 2545   E.wrex 2546   _Vcvv 2790    C_ wss 3154   (/)c0 3457   class class class wbr 4025    e. cmpt 4079    _E cep 4305    Fr wfr 4351   suc csuc 4396   omcom 4658   dom cdm 4691   ran crn 4692    Fn wfn 5252   ` cfv 5257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4216  ax-un 4514  ax-reg 7308  ax-inf2 7344
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265
  Copyright terms: Public domain W3C validator