MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inf3lem6 Unicode version

Theorem inf3lem6 7334
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7336 for detailed description. (Contributed by NM, 29-Oct-1996.)
Hypotheses
Ref Expression
inf3lem.1  |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
inf3lem.2  |-  F  =  ( rec ( G ,  (/) )  |`  om )
inf3lem.3  |-  A  e. 
_V
inf3lem.4  |-  B  e. 
_V
Assertion
Ref Expression
inf3lem6  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  F : om -1-1-> ~P x
)
Distinct variable group:    x, y, w
Allowed substitution hints:    A( x, y, w)    B( x, y, w)    F( x, y, w)    G( x, y, w)

Proof of Theorem inf3lem6
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inf3lem.1 . . . . . . . . . . 11  |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
2 inf3lem.2 . . . . . . . . . . 11  |-  F  =  ( rec ( G ,  (/) )  |`  om )
3 vex 2791 . . . . . . . . . . 11  |-  u  e. 
_V
4 vex 2791 . . . . . . . . . . 11  |-  v  e. 
_V
51, 2, 3, 4inf3lem5 7333 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( u  e. 
om  /\  v  e.  u )  ->  ( F `  v )  C.  ( F `  u
) ) )
6 dfpss2 3261 . . . . . . . . . . 11  |-  ( ( F `  v ) 
C.  ( F `  u )  <->  ( ( F `  v )  C_  ( F `  u
)  /\  -.  ( F `  v )  =  ( F `  u ) ) )
76simprbi 450 . . . . . . . . . 10  |-  ( ( F `  v ) 
C.  ( F `  u )  ->  -.  ( F `  v )  =  ( F `  u ) )
85, 7syl6 29 . . . . . . . . 9  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( u  e. 
om  /\  v  e.  u )  ->  -.  ( F `  v )  =  ( F `  u ) ) )
98expdimp 426 . . . . . . . 8  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  u  e.  om )  ->  ( v  e.  u  ->  -.  ( F `  v )  =  ( F `  u ) ) )
109adantrl 696 . . . . . . 7  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( v  e.  u  ->  -.  ( F `  v )  =  ( F `  u ) ) )
111, 2, 4, 3inf3lem5 7333 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( v  e. 
om  /\  u  e.  v )  ->  ( F `  u )  C.  ( F `  v
) ) )
12 dfpss2 3261 . . . . . . . . . . . 12  |-  ( ( F `  u ) 
C.  ( F `  v )  <->  ( ( F `  u )  C_  ( F `  v
)  /\  -.  ( F `  u )  =  ( F `  v ) ) )
1312simprbi 450 . . . . . . . . . . 11  |-  ( ( F `  u ) 
C.  ( F `  v )  ->  -.  ( F `  u )  =  ( F `  v ) )
14 eqcom 2285 . . . . . . . . . . 11  |-  ( ( F `  u )  =  ( F `  v )  <->  ( F `  v )  =  ( F `  u ) )
1513, 14sylnib 295 . . . . . . . . . 10  |-  ( ( F `  u ) 
C.  ( F `  v )  ->  -.  ( F `  v )  =  ( F `  u ) )
1611, 15syl6 29 . . . . . . . . 9  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( v  e. 
om  /\  u  e.  v )  ->  -.  ( F `  v )  =  ( F `  u ) ) )
1716expdimp 426 . . . . . . . 8  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  v  e.  om )  ->  ( u  e.  v  ->  -.  ( F `  v )  =  ( F `  u ) ) )
1817adantrr 697 . . . . . . 7  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( u  e.  v  ->  -.  ( F `  v )  =  ( F `  u ) ) )
1910, 18jaod 369 . . . . . 6  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( (
v  e.  u  \/  u  e.  v )  ->  -.  ( F `  v )  =  ( F `  u ) ) )
2019con2d 107 . . . . 5  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( ( F `  v )  =  ( F `  u )  ->  -.  ( v  e.  u  \/  u  e.  v
) ) )
21 nnord 4664 . . . . . . 7  |-  ( v  e.  om  ->  Ord  v )
22 nnord 4664 . . . . . . 7  |-  ( u  e.  om  ->  Ord  u )
23 ordtri3 4428 . . . . . . 7  |-  ( ( Ord  v  /\  Ord  u )  ->  (
v  =  u  <->  -.  (
v  e.  u  \/  u  e.  v ) ) )
2421, 22, 23syl2an 463 . . . . . 6  |-  ( ( v  e.  om  /\  u  e.  om )  ->  ( v  =  u  <->  -.  ( v  e.  u  \/  u  e.  v
) ) )
2524adantl 452 . . . . 5  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( v  =  u  <->  -.  ( v  e.  u  \/  u  e.  v ) ) )
2620, 25sylibrd 225 . . . 4  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( ( F `  v )  =  ( F `  u )  ->  v  =  u ) )
2726ralrimivva 2635 . . 3  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  A. v  e.  om  A. u  e.  om  (
( F `  v
)  =  ( F `
 u )  -> 
v  =  u ) )
28 frfnom 6447 . . . . . 6  |-  ( rec ( G ,  (/) )  |`  om )  Fn 
om
29 fneq1 5333 . . . . . 6  |-  ( F  =  ( rec ( G ,  (/) )  |`  om )  ->  ( F  Fn  om  <->  ( rec ( G ,  (/) )  |`  om )  Fn  om )
)
3028, 29mpbiri 224 . . . . 5  |-  ( F  =  ( rec ( G ,  (/) )  |`  om )  ->  F  Fn  om )
31 fvelrnb 5570 . . . . . . . 8  |-  ( F  Fn  om  ->  (
u  e.  ran  F  <->  E. v  e.  om  ( F `  v )  =  u ) )
32 inf3lem.4 . . . . . . . . . . . 12  |-  B  e. 
_V
331, 2, 4, 32inf3lemd 7328 . . . . . . . . . . 11  |-  ( v  e.  om  ->  ( F `  v )  C_  x )
34 fvex 5539 . . . . . . . . . . . 12  |-  ( F `
 v )  e. 
_V
3534elpw 3631 . . . . . . . . . . 11  |-  ( ( F `  v )  e.  ~P x  <->  ( F `  v )  C_  x
)
3633, 35sylibr 203 . . . . . . . . . 10  |-  ( v  e.  om  ->  ( F `  v )  e.  ~P x )
37 eleq1 2343 . . . . . . . . . 10  |-  ( ( F `  v )  =  u  ->  (
( F `  v
)  e.  ~P x  <->  u  e.  ~P x ) )
3836, 37syl5ibcom 211 . . . . . . . . 9  |-  ( v  e.  om  ->  (
( F `  v
)  =  u  ->  u  e.  ~P x
) )
3938rexlimiv 2661 . . . . . . . 8  |-  ( E. v  e.  om  ( F `  v )  =  u  ->  u  e. 
~P x )
4031, 39syl6bi 219 . . . . . . 7  |-  ( F  Fn  om  ->  (
u  e.  ran  F  ->  u  e.  ~P x
) )
4140ssrdv 3185 . . . . . 6  |-  ( F  Fn  om  ->  ran  F 
C_  ~P x )
4241ancli 534 . . . . 5  |-  ( F  Fn  om  ->  ( F  Fn  om  /\  ran  F 
C_  ~P x ) )
432, 30, 42mp2b 9 . . . 4  |-  ( F  Fn  om  /\  ran  F 
C_  ~P x )
44 df-f 5259 . . . 4  |-  ( F : om --> ~P x  <->  ( F  Fn  om  /\  ran  F  C_  ~P x
) )
4543, 44mpbir 200 . . 3  |-  F : om
--> ~P x
4627, 45jctil 523 . 2  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F : om --> ~P x  /\  A. v  e.  om  A. u  e. 
om  ( ( F `
 v )  =  ( F `  u
)  ->  v  =  u ) ) )
47 dff13 5783 . 2  |-  ( F : om -1-1-> ~P x  <->  ( F : om --> ~P x  /\  A. v  e.  om  A. u  e.  om  (
( F `  v
)  =  ( F `
 u )  -> 
v  =  u ) ) )
4846, 47sylibr 203 1  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  F : om -1-1-> ~P x
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152    C. wpss 3153   (/)c0 3455   ~Pcpw 3625   U.cuni 3827    e. cmpt 4077   Ord word 4391   omcom 4656   ran crn 4690    |` cres 4691    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   ` cfv 5255   reccrdg 6422
This theorem is referenced by:  inf3lem7  7335  dominf  8071  dominfac  8195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423
  Copyright terms: Public domain W3C validator