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Theorem inf3lem6 7839
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7841 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 2975 . . . . . . . . . . 11  |-  u  e. 
_V
4 vex 2975 . . . . . . . . . . 11  |-  v  e. 
_V
51, 2, 3, 4inf3lem5 7838 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( u  e. 
om  /\  v  e.  u )  ->  ( F `  v )  C.  ( F `  u
) ) )
6 dfpss2 3441 . . . . . . . . . . 11  |-  ( ( F `  v ) 
C.  ( F `  u )  <->  ( ( F `  v )  C_  ( F `  u
)  /\  -.  ( F `  v )  =  ( F `  u ) ) )
76simprbi 464 . . . . . . . . . 10  |-  ( ( F `  v ) 
C.  ( F `  u )  ->  -.  ( F `  v )  =  ( F `  u ) )
85, 7syl6 33 . . . . . . . . 9  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( u  e. 
om  /\  v  e.  u )  ->  -.  ( F `  v )  =  ( F `  u ) ) )
98expdimp 437 . . . . . . . 8  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  u  e.  om )  ->  ( v  e.  u  ->  -.  ( F `  v )  =  ( F `  u ) ) )
109adantrl 715 . . . . . . 7  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( v  e.  u  ->  -.  ( F `  v )  =  ( F `  u ) ) )
111, 2, 4, 3inf3lem5 7838 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( v  e. 
om  /\  u  e.  v )  ->  ( F `  u )  C.  ( F `  v
) ) )
12 dfpss2 3441 . . . . . . . . . . . 12  |-  ( ( F `  u ) 
C.  ( F `  v )  <->  ( ( F `  u )  C_  ( F `  v
)  /\  -.  ( F `  u )  =  ( F `  v ) ) )
1312simprbi 464 . . . . . . . . . . 11  |-  ( ( F `  u ) 
C.  ( F `  v )  ->  -.  ( F `  u )  =  ( F `  v ) )
14 eqcom 2445 . . . . . . . . . . 11  |-  ( ( F `  u )  =  ( F `  v )  <->  ( F `  v )  =  ( F `  u ) )
1513, 14sylnib 304 . . . . . . . . . 10  |-  ( ( F `  u ) 
C.  ( F `  v )  ->  -.  ( F `  v )  =  ( F `  u ) )
1611, 15syl6 33 . . . . . . . . 9  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( v  e. 
om  /\  u  e.  v )  ->  -.  ( F `  v )  =  ( F `  u ) ) )
1716expdimp 437 . . . . . . . 8  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  v  e.  om )  ->  ( u  e.  v  ->  -.  ( F `  v )  =  ( F `  u ) ) )
1817adantrr 716 . . . . . . 7  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( u  e.  v  ->  -.  ( F `  v )  =  ( F `  u ) ) )
1910, 18jaod 380 . . . . . 6  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( (
v  e.  u  \/  u  e.  v )  ->  -.  ( F `  v )  =  ( F `  u ) ) )
2019con2d 115 . . . . 5  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( ( F `  v )  =  ( F `  u )  ->  -.  ( v  e.  u  \/  u  e.  v
) ) )
21 nnord 6484 . . . . . . 7  |-  ( v  e.  om  ->  Ord  v )
22 nnord 6484 . . . . . . 7  |-  ( u  e.  om  ->  Ord  u )
23 ordtri3 4755 . . . . . . 7  |-  ( ( Ord  v  /\  Ord  u )  ->  (
v  =  u  <->  -.  (
v  e.  u  \/  u  e.  v ) ) )
2421, 22, 23syl2an 477 . . . . . 6  |-  ( ( v  e.  om  /\  u  e.  om )  ->  ( v  =  u  <->  -.  ( v  e.  u  \/  u  e.  v
) ) )
2524adantl 466 . . . . 5  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( v  =  u  <->  -.  ( v  e.  u  \/  u  e.  v ) ) )
2620, 25sylibrd 234 . . . 4  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( ( F `  v )  =  ( F `  u )  ->  v  =  u ) )
2726ralrimivva 2808 . . 3  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  A. v  e.  om  A. u  e.  om  (
( F `  v
)  =  ( F `
 u )  -> 
v  =  u ) )
28 frfnom 6890 . . . . . 6  |-  ( rec ( G ,  (/) )  |`  om )  Fn 
om
29 fneq1 5499 . . . . . 6  |-  ( F  =  ( rec ( G ,  (/) )  |`  om )  ->  ( F  Fn  om  <->  ( rec ( G ,  (/) )  |`  om )  Fn  om )
)
3028, 29mpbiri 233 . . . . 5  |-  ( F  =  ( rec ( G ,  (/) )  |`  om )  ->  F  Fn  om )
31 fvelrnb 5739 . . . . . . . 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 7833 . . . . . . . . . . 11  |-  ( v  e.  om  ->  ( F `  v )  C_  x )
34 fvex 5701 . . . . . . . . . . . 12  |-  ( F `
 v )  e. 
_V
3534elpw 3866 . . . . . . . . . . 11  |-  ( ( F `  v )  e.  ~P x  <->  ( F `  v )  C_  x
)
3633, 35sylibr 212 . . . . . . . . . 10  |-  ( v  e.  om  ->  ( F `  v )  e.  ~P x )
37 eleq1 2503 . . . . . . . . . 10  |-  ( ( F `  v )  =  u  ->  (
( F `  v
)  e.  ~P x  <->  u  e.  ~P x ) )
3836, 37syl5ibcom 220 . . . . . . . . 9  |-  ( v  e.  om  ->  (
( F `  v
)  =  u  ->  u  e.  ~P x
) )
3938rexlimiv 2835 . . . . . . . 8  |-  ( E. v  e.  om  ( F `  v )  =  u  ->  u  e. 
~P x )
4031, 39syl6bi 228 . . . . . . 7  |-  ( F  Fn  om  ->  (
u  e.  ran  F  ->  u  e.  ~P x
) )
4140ssrdv 3362 . . . . . 6  |-  ( F  Fn  om  ->  ran  F 
C_  ~P x )
4241ancli 551 . . . . 5  |-  ( F  Fn  om  ->  ( F  Fn  om  /\  ran  F 
C_  ~P x ) )
432, 30, 42mp2b 10 . . . 4  |-  ( F  Fn  om  /\  ran  F 
C_  ~P x )
44 df-f 5422 . . . 4  |-  ( F : om --> ~P x  <->  ( F  Fn  om  /\  ran  F  C_  ~P x
) )
4543, 44mpbir 209 . . 3  |-  F : om
--> ~P x
4627, 45jctil 537 . 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 5971 . 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 212 1  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  F : om -1-1-> ~P x
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   {crab 2719   _Vcvv 2972    i^i cin 3327    C_ wss 3328    C. wpss 3329   (/)c0 3637   ~Pcpw 3860   U.cuni 4091    e. cmpt 4350   Ord word 4718   ran crn 4841    |` cres 4842    Fn wfn 5413   -->wf 5414   -1-1->wf1 5415   ` cfv 5418   omcom 6476   reccrdg 6865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-reg 7807
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-om 6477  df-recs 6832  df-rdg 6866
This theorem is referenced by:  inf3lem7  7840  dominf  8614  dominfac  8737
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