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Theorem inf3lem6 8135
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8137 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 3047 . . . . . . . . . . 11  |-  u  e. 
_V
4 vex 3047 . . . . . . . . . . 11  |-  v  e. 
_V
51, 2, 3, 4inf3lem5 8134 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( u  e. 
om  /\  v  e.  u )  ->  ( F `  v )  C.  ( F `  u
) ) )
6 dfpss2 3517 . . . . . . . . . . 11  |-  ( ( F `  v ) 
C.  ( F `  u )  <->  ( ( F `  v )  C_  ( F `  u
)  /\  -.  ( F `  v )  =  ( F `  u ) ) )
76simprbi 466 . . . . . . . . . 10  |-  ( ( F `  v ) 
C.  ( F `  u )  ->  -.  ( F `  v )  =  ( F `  u ) )
85, 7syl6 34 . . . . . . . . 9  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( u  e. 
om  /\  v  e.  u )  ->  -.  ( F `  v )  =  ( F `  u ) ) )
98expdimp 439 . . . . . . . 8  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  u  e.  om )  ->  ( v  e.  u  ->  -.  ( F `  v )  =  ( F `  u ) ) )
109adantrl 721 . . . . . . 7  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( v  e.  u  ->  -.  ( F `  v )  =  ( F `  u ) ) )
111, 2, 4, 3inf3lem5 8134 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( v  e. 
om  /\  u  e.  v )  ->  ( F `  u )  C.  ( F `  v
) ) )
12 dfpss2 3517 . . . . . . . . . . . 12  |-  ( ( F `  u ) 
C.  ( F `  v )  <->  ( ( F `  u )  C_  ( F `  v
)  /\  -.  ( F `  u )  =  ( F `  v ) ) )
1312simprbi 466 . . . . . . . . . . 11  |-  ( ( F `  u ) 
C.  ( F `  v )  ->  -.  ( F `  u )  =  ( F `  v ) )
14 eqcom 2457 . . . . . . . . . . 11  |-  ( ( F `  u )  =  ( F `  v )  <->  ( F `  v )  =  ( F `  u ) )
1513, 14sylnib 306 . . . . . . . . . 10  |-  ( ( F `  u ) 
C.  ( F `  v )  ->  -.  ( F `  v )  =  ( F `  u ) )
1611, 15syl6 34 . . . . . . . . 9  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( v  e. 
om  /\  u  e.  v )  ->  -.  ( F `  v )  =  ( F `  u ) ) )
1716expdimp 439 . . . . . . . 8  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  v  e.  om )  ->  ( u  e.  v  ->  -.  ( F `  v )  =  ( F `  u ) ) )
1817adantrr 722 . . . . . . 7  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( u  e.  v  ->  -.  ( F `  v )  =  ( F `  u ) ) )
1910, 18jaod 382 . . . . . 6  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( (
v  e.  u  \/  u  e.  v )  ->  -.  ( F `  v )  =  ( F `  u ) ) )
2019con2d 119 . . . . 5  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( ( F `  v )  =  ( F `  u )  ->  -.  ( v  e.  u  \/  u  e.  v
) ) )
21 nnord 6697 . . . . . . 7  |-  ( v  e.  om  ->  Ord  v )
22 nnord 6697 . . . . . . 7  |-  ( u  e.  om  ->  Ord  u )
23 ordtri3 5458 . . . . . . 7  |-  ( ( Ord  v  /\  Ord  u )  ->  (
v  =  u  <->  -.  (
v  e.  u  \/  u  e.  v ) ) )
2421, 22, 23syl2an 480 . . . . . 6  |-  ( ( v  e.  om  /\  u  e.  om )  ->  ( v  =  u  <->  -.  ( v  e.  u  \/  u  e.  v
) ) )
2524adantl 468 . . . . 5  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( v  =  u  <->  -.  ( v  e.  u  \/  u  e.  v ) ) )
2620, 25sylibrd 238 . . . 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 7149 . . . . . 6  |-  ( rec ( G ,  (/) )  |`  om )  Fn 
om
29 fneq1 5662 . . . . . 6  |-  ( F  =  ( rec ( G ,  (/) )  |`  om )  ->  ( F  Fn  om  <->  ( rec ( G ,  (/) )  |`  om )  Fn  om )
)
3028, 29mpbiri 237 . . . . 5  |-  ( F  =  ( rec ( G ,  (/) )  |`  om )  ->  F  Fn  om )
31 fvelrnb 5910 . . . . . . . 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 8129 . . . . . . . . . . 11  |-  ( v  e.  om  ->  ( F `  v )  C_  x )
34 fvex 5873 . . . . . . . . . . . 12  |-  ( F `
 v )  e. 
_V
3534elpw 3956 . . . . . . . . . . 11  |-  ( ( F `  v )  e.  ~P x  <->  ( F `  v )  C_  x
)
3633, 35sylibr 216 . . . . . . . . . 10  |-  ( v  e.  om  ->  ( F `  v )  e.  ~P x )
37 eleq1 2516 . . . . . . . . . 10  |-  ( ( F `  v )  =  u  ->  (
( F `  v
)  e.  ~P x  <->  u  e.  ~P x ) )
3836, 37syl5ibcom 224 . . . . . . . . 9  |-  ( v  e.  om  ->  (
( F `  v
)  =  u  ->  u  e.  ~P x
) )
3938rexlimiv 2872 . . . . . . . 8  |-  ( E. v  e.  om  ( F `  v )  =  u  ->  u  e. 
~P x )
4031, 39syl6bi 232 . . . . . . 7  |-  ( F  Fn  om  ->  (
u  e.  ran  F  ->  u  e.  ~P x
) )
4140ssrdv 3437 . . . . . 6  |-  ( F  Fn  om  ->  ran  F 
C_  ~P x )
4241ancli 554 . . . . 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 5585 . . . 4  |-  ( F : om --> ~P x  <->  ( F  Fn  om  /\  ran  F  C_  ~P x
) )
4543, 44mpbir 213 . . 3  |-  F : om
--> ~P x
4627, 45jctil 540 . 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 6157 . 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 216 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 188    \/ wo 370    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 3044    i^i cin 3402    C_ wss 3403    C. wpss 3404   (/)c0 3730   ~Pcpw 3950   U.cuni 4197    |-> cmpt 4460   ran crn 4834    |` cres 4835   Ord word 5421    Fn wfn 5576   -->wf 5577   -1-1->wf1 5578   ` cfv 5581   omcom 6689   reccrdg 7124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-reg 8104
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6690  df-wrecs 7025  df-recs 7087  df-rdg 7125
This theorem is referenced by:  inf3lem7  8136  dominf  8872  dominfac  8995
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