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Theorem noinfep 7607
Description: Using the Axiom of Regularity in the form zfregfr 7563, 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 7591 . . . . 5  |-  om  e.  _V
21mptex 5959 . . . 4  |-  ( w  e.  om  |->  ( F `
 w ) )  e.  _V
32rnex 5126 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  e.  _V
4 zfregfr 7563 . . 3  |-  _E  Fr  ran  ( w  e.  om  |->  ( F `  w ) )
5 ssid 3360 . . 3  |-  ran  (
w  e.  om  |->  ( F `  w ) )  C_  ran  ( w  e.  om  |->  ( F `
 w ) )
6 dmmptg 5360 . . . . . 6  |-  ( A. w  e.  om  ( F `  w )  e.  _V  ->  dom  ( w  e.  om  |->  ( F `
 w ) )  =  om )
7 fvex 5735 . . . . . . 7  |-  ( F `
 w )  e. 
_V
87a1i 11 . . . . . 6  |-  ( w  e.  om  ->  ( F `  w )  e.  _V )
96, 8mprg 2768 . . . . 5  |-  dom  (
w  e.  om  |->  ( F `  w ) )  =  om
10 peano1 4857 . . . . . 6  |-  (/)  e.  om
11 ne0i 3627 . . . . . 6  |-  ( (/)  e.  om  ->  om  =/=  (/) )
1210, 11ax-mp 8 . . . . 5  |-  om  =/=  (/)
139, 12eqnetri 2616 . . . 4  |-  dom  (
w  e.  om  |->  ( F `  w ) )  =/=  (/)
14 dm0rn0 5079 . . . . 5  |-  ( dom  ( w  e.  om  |->  ( F `  w ) )  =  (/)  <->  ran  ( w  e.  om  |->  ( F `
 w ) )  =  (/) )
1514necon3bii 2631 . . . 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 4537 . . 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 2436 . . . . . . 7  |-  ( w  e.  om  |->  ( F `
 w ) )  =  ( w  e. 
om  |->  ( F `  w ) )
207, 19fnmpti 5566 . . . . . 6  |-  ( w  e.  om  |->  ( F `
 w ) )  Fn  om
21 fvelrnb 5767 . . . . . 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 4858 . . . . . . . . . . 11  |-  ( x  e.  om  ->  suc  x  e.  om )
24 fveq2 5721 . . . . . . . . . . . 12  |-  ( w  =  suc  x  -> 
( F `  w
)  =  ( F `
 suc  x )
)
25 fvex 5735 . . . . . . . . . . . 12  |-  ( F `
 suc  x )  e.  _V
2624, 19, 25fvmpt 5799 . . . . . . . . . . 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 5860 . . . . . . . . . . 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 2511 . . . . . . . . 9  |-  ( x  e.  om  ->  ( F `  suc  x )  e.  ran  ( w  e.  om  |->  ( F `
 w ) ) )
31 epel 4490 . . . . . . . . . . . . 13  |-  ( z  _E  y  <->  z  e.  y )
32 eleq1 2496 . . . . . . . . . . . . 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 2602 . . . . . . . . . . 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 3042 . . . . . . . . 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 5721 . . . . . . . . . . . 12  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
40 fvex 5735 . . . . . . . . . . . 12  |-  ( F `
 x )  e. 
_V
4139, 19, 40fvmpt 5799 . . . . . . . . . . 11  |-  ( x  e.  om  ->  (
( w  e.  om  |->  ( F `  w ) ) `  x )  =  ( F `  x ) )
42 eqeq1 2442 . . . . . . . . . . 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 2693 . . . . . . . . . . 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 2809 . . . . 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 2817 . 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 1652    e. wcel 1725    =/= wne 2599    e/ wnel 2600   A.wral 2698   E.wrex 2699   _Vcvv 2949    C_ wss 3313   (/)c0 3621   class class class wbr 4205    e. cmpt 4259    _E cep 4485    Fr wfr 4531   suc csuc 4576   omcom 4838   dom cdm 4871   ran crn 4872    Fn wfn 5442   ` cfv 5447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pr 4396  ax-un 4694  ax-reg 7553  ax-inf2 7589
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455
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