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

Theorem noinfepOLD 7571
Description: Using the Axiom of Regularity in the form zfregfr 7526, show that there are no infinite descending 
e.-chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
noinfepOLD  |-  ( F  Fn  om  ->  E. x  e.  om  -.  ( F `
 suc  x )  e.  ( F `  x
) )
Distinct variable group:    x, F

Proof of Theorem noinfepOLD
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fndm 5503 . . . . 5  |-  ( F  Fn  om  ->  dom  F  =  om )
2 omex 7554 . . . . 5  |-  om  e.  _V
31, 2syl6eqel 2492 . . . 4  |-  ( F  Fn  om  ->  dom  F  e.  _V )
4 fnfun 5501 . . . 4  |-  ( F  Fn  om  ->  Fun  F )
5 funrnex 5926 . . . 4  |-  ( dom 
F  e.  _V  ->  ( Fun  F  ->  ran  F  e.  _V ) )
63, 4, 5sylc 58 . . 3  |-  ( F  Fn  om  ->  ran  F  e.  _V )
7 peano1 4823 . . . . . . 7  |-  (/)  e.  om
8 eleq2 2465 . . . . . . 7  |-  ( dom 
F  =  om  ->  (
(/)  e.  dom  F  <->  (/)  e.  om ) )
97, 8mpbiri 225 . . . . . 6  |-  ( dom 
F  =  om  ->  (/)  e.  dom  F )
10 ne0i 3594 . . . . . 6  |-  ( (/)  e.  dom  F  ->  dom  F  =/=  (/) )
119, 10syl 16 . . . . 5  |-  ( dom 
F  =  om  ->  dom 
F  =/=  (/) )
12 dm0rn0 5045 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1312necon3bii 2599 . . . . 5  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
1411, 13sylib 189 . . . 4  |-  ( dom 
F  =  om  ->  ran 
F  =/=  (/) )
151, 14syl 16 . . 3  |-  ( F  Fn  om  ->  ran  F  =/=  (/) )
16 zfregfr 7526 . . . 4  |-  _E  Fr  ran  F
17 ssid 3327 . . . . 5  |-  ran  F  C_ 
ran  F
18 fri 4504 . . . . 5  |-  ( ( ( ran  F  e. 
_V  /\  _E  Fr  ran  F )  /\  ( ran  F  C_  ran  F  /\  ran  F  =/=  (/) ) )  ->  E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y
)
1917, 18mpanr1 665 . . . 4  |-  ( ( ( ran  F  e. 
_V  /\  _E  Fr  ran  F )  /\  ran  F  =/=  (/) )  ->  E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y )
2016, 19mpanl2 663 . . 3  |-  ( ( ran  F  e.  _V  /\ 
ran  F  =/=  (/) )  ->  E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y
)
216, 15, 20syl2anc 643 . 2  |-  ( F  Fn  om  ->  E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y )
22 fvelrnb 5733 . . . . . . 7  |-  ( F  Fn  om  ->  (
y  e.  ran  F  <->  E. x  e.  om  ( F `  x )  =  y ) )
2322adantr 452 . . . . . 6  |-  ( ( F  Fn  om  /\  A. z  e.  ran  F  -.  z  _E  y
)  ->  ( y  e.  ran  F  <->  E. x  e.  om  ( F `  x )  =  y ) )
24 peano2 4824 . . . . . . . . 9  |-  ( x  e.  om  ->  suc  x  e.  om )
25 fnfvelrn 5826 . . . . . . . . . . . 12  |-  ( ( F  Fn  om  /\  suc  x  e.  om )  ->  ( F `  suc  x )  e.  ran  F )
2625adantlr 696 . . . . . . . . . . 11  |-  ( ( ( F  Fn  om  /\ 
A. z  e.  ran  F  -.  z  _E  y
)  /\  suc  x  e. 
om )  ->  ( F `  suc  x )  e.  ran  F )
27 simplr 732 . . . . . . . . . . 11  |-  ( ( ( F  Fn  om  /\ 
A. z  e.  ran  F  -.  z  _E  y
)  /\  suc  x  e. 
om )  ->  A. z  e.  ran  F  -.  z  _E  y )
2826, 27jca 519 . . . . . . . . . 10  |-  ( ( ( F  Fn  om  /\ 
A. z  e.  ran  F  -.  z  _E  y
)  /\  suc  x  e. 
om )  ->  (
( F `  suc  x )  e.  ran  F  /\  A. z  e. 
ran  F  -.  z  _E  y ) )
29 epel 4457 . . . . . . . . . . . . . 14  |-  ( z  _E  y  <->  z  e.  y )
30 eleq1 2464 . . . . . . . . . . . . . 14  |-  ( z  =  ( F `  suc  x )  ->  (
z  e.  y  <->  ( F `  suc  x )  e.  y ) )
3129, 30syl5bb 249 . . . . . . . . . . . . 13  |-  ( z  =  ( F `  suc  x )  ->  (
z  _E  y  <->  ( F `  suc  x )  e.  y ) )
3231notbid 286 . . . . . . . . . . . 12  |-  ( z  =  ( F `  suc  x )  ->  ( -.  z  _E  y  <->  -.  ( F `  suc  x )  e.  y ) )
3332rspcva 3010 . . . . . . . . . . 11  |-  ( ( ( F `  suc  x )  e.  ran  F  /\  A. z  e. 
ran  F  -.  z  _E  y )  ->  -.  ( F `  suc  x
)  e.  y )
34 eleq2 2465 . . . . . . . . . . . 12  |-  ( ( F `  x )  =  y  ->  (
( F `  suc  x )  e.  ( F `  x )  <-> 
( F `  suc  x )  e.  y ) )
3534notbid 286 . . . . . . . . . . 11  |-  ( ( F `  x )  =  y  ->  ( -.  ( F `  suc  x )  e.  ( F `  x )  <->  -.  ( F `  suc  x )  e.  y ) )
3633, 35syl5ibr 213 . . . . . . . . . 10  |-  ( ( F `  x )  =  y  ->  (
( ( F `  suc  x )  e.  ran  F  /\  A. z  e. 
ran  F  -.  z  _E  y )  ->  -.  ( F `  suc  x
)  e.  ( F `
 x ) ) )
3728, 36syl5 30 . . . . . . . . 9  |-  ( ( F `  x )  =  y  ->  (
( ( F  Fn  om 
/\  A. z  e.  ran  F  -.  z  _E  y
)  /\  suc  x  e. 
om )  ->  -.  ( F `  suc  x
)  e.  ( F `
 x ) ) )
3824, 37sylan2i 637 . . . . . . . 8  |-  ( ( F `  x )  =  y  ->  (
( ( F  Fn  om 
/\  A. z  e.  ran  F  -.  z  _E  y
)  /\  x  e.  om )  ->  -.  ( F `  suc  x )  e.  ( F `  x ) ) )
3938com12 29 . . . . . . 7  |-  ( ( ( F  Fn  om  /\ 
A. z  e.  ran  F  -.  z  _E  y
)  /\  x  e.  om )  ->  ( ( F `  x )  =  y  ->  -.  ( F `  suc  x )  e.  ( F `  x ) ) )
4039reximdva 2778 . . . . . 6  |-  ( ( F  Fn  om  /\  A. z  e.  ran  F  -.  z  _E  y
)  ->  ( E. x  e.  om  ( F `  x )  =  y  ->  E. x  e.  om  -.  ( F `
 suc  x )  e.  ( F `  x
) ) )
4123, 40sylbid 207 . . . . 5  |-  ( ( F  Fn  om  /\  A. z  e.  ran  F  -.  z  _E  y
)  ->  ( y  e.  ran  F  ->  E. x  e.  om  -.  ( F `
 suc  x )  e.  ( F `  x
) ) )
4241ex 424 . . . 4  |-  ( F  Fn  om  ->  ( A. z  e.  ran  F  -.  z  _E  y  ->  ( y  e.  ran  F  ->  E. x  e.  om  -.  ( F `  suc  x )  e.  ( F `  x ) ) ) )
4342com23 74 . . 3  |-  ( F  Fn  om  ->  (
y  e.  ran  F  ->  ( A. z  e. 
ran  F  -.  z  _E  y  ->  E. x  e.  om  -.  ( F `
 suc  x )  e.  ( F `  x
) ) ) )
4443rexlimdv 2789 . 2  |-  ( F  Fn  om  ->  ( E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y  ->  E. x  e.  om  -.  ( F `  suc  x )  e.  ( F `  x ) ) )
4521, 44mpd 15 1  |-  ( F  Fn  om  ->  E. x  e.  om  -.  ( F `
 suc  x )  e.  ( F `  x
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   _Vcvv 2916    C_ wss 3280   (/)c0 3588   class class class wbr 4172    _E cep 4452    Fr wfr 4498   suc csuc 4543   omcom 4804   dom cdm 4837   ran crn 4838   Fun wfun 5407    Fn wfn 5408   ` cfv 5413
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660  ax-reg 7516  ax-inf2 7552
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421
  Copyright terms: Public domain W3C validator