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Theorem inf3lem6 8039
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8041 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 3109 . . . . . . . . . . 11  |-  u  e. 
_V
4 vex 3109 . . . . . . . . . . 11  |-  v  e. 
_V
51, 2, 3, 4inf3lem5 8038 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( u  e. 
om  /\  v  e.  u )  ->  ( F `  v )  C.  ( F `  u
) ) )
6 dfpss2 3582 . . . . . . . . . . 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 8038 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( v  e. 
om  /\  u  e.  v )  ->  ( F `  u )  C.  ( F `  v
) ) )
12 dfpss2 3582 . . . . . . . . . . . 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 2469 . . . . . . . . . . 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 6679 . . . . . . 7  |-  ( v  e.  om  ->  Ord  v )
22 nnord 6679 . . . . . . 7  |-  ( u  e.  om  ->  Ord  u )
23 ordtri3 4907 . . . . . . 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 2878 . . 3  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  A. v  e.  om  A. u  e.  om  (
( F `  v
)  =  ( F `
 u )  -> 
v  =  u ) )
28 frfnom 7090 . . . . . 6  |-  ( rec ( G ,  (/) )  |`  om )  Fn 
om
29 fneq1 5660 . . . . . 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 5906 . . . . . . . 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 8033 . . . . . . . . . . 11  |-  ( v  e.  om  ->  ( F `  v )  C_  x )
34 fvex 5867 . . . . . . . . . . . 12  |-  ( F `
 v )  e. 
_V
3534elpw 4009 . . . . . . . . . . 11  |-  ( ( F `  v )  e.  ~P x  <->  ( F `  v )  C_  x
)
3633, 35sylibr 212 . . . . . . . . . 10  |-  ( v  e.  om  ->  ( F `  v )  e.  ~P x )
37 eleq1 2532 . . . . . . . . . 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 2942 . . . . . . . 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 3503 . . . . . 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 5583 . . . 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 6145 . 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 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3106    i^i cin 3468    C_ wss 3469    C. wpss 3470   (/)c0 3778   ~Pcpw 4003   U.cuni 4238    |-> cmpt 4498   Ord word 4870   ran crn 4993    |` cres 4994    Fn wfn 5574   -->wf 5575   -1-1->wf1 5576   ` cfv 5579   omcom 6671   reccrdg 7065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-reg 8007
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-recs 7032  df-rdg 7066
This theorem is referenced by:  inf3lem7  8040  dominf  8814  dominfac  8937
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