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Theorem inf3lem6 8085
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8087 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 3064 . . . . . . . . . . 11  |-  u  e. 
_V
4 vex 3064 . . . . . . . . . . 11  |-  v  e. 
_V
51, 2, 3, 4inf3lem5 8084 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( u  e. 
om  /\  v  e.  u )  ->  ( F `  v )  C.  ( F `  u
) ) )
6 dfpss2 3530 . . . . . . . . . . 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 716 . . . . . . 7  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( v  e.  u  ->  -.  ( F `  v )  =  ( F `  u ) ) )
111, 2, 4, 3inf3lem5 8084 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( v  e. 
om  /\  u  e.  v )  ->  ( F `  u )  C.  ( F `  v
) ) )
12 dfpss2 3530 . . . . . . . . . . . 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 2413 . . . . . . . . . . 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 717 . . . . . . 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 117 . . . . 5  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( ( F `  v )  =  ( F `  u )  ->  -.  ( v  e.  u  \/  u  e.  v
) ) )
21 nnord 6693 . . . . . . 7  |-  ( v  e.  om  ->  Ord  v )
22 nnord 6693 . . . . . . 7  |-  ( u  e.  om  ->  Ord  u )
23 ordtri3 5448 . . . . . . 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 236 . . . 4  |-  ( ( ( x  =/=  (/)  /\  x  C_ 
U. x )  /\  ( v  e.  om  /\  u  e.  om )
)  ->  ( ( F `  v )  =  ( F `  u )  ->  v  =  u ) )
2726ralrimivva 2827 . . 3  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  A. v  e.  om  A. u  e.  om  (
( F `  v
)  =  ( F `
 u )  -> 
v  =  u ) )
28 frfnom 7139 . . . . . 6  |-  ( rec ( G ,  (/) )  |`  om )  Fn 
om
29 fneq1 5652 . . . . . 6  |-  ( F  =  ( rec ( G ,  (/) )  |`  om )  ->  ( F  Fn  om  <->  ( rec ( G ,  (/) )  |`  om )  Fn  om )
)
3028, 29mpbiri 235 . . . . 5  |-  ( F  =  ( rec ( G ,  (/) )  |`  om )  ->  F  Fn  om )
31 fvelrnb 5898 . . . . . . . 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 8079 . . . . . . . . . . 11  |-  ( v  e.  om  ->  ( F `  v )  C_  x )
34 fvex 5861 . . . . . . . . . . . 12  |-  ( F `
 v )  e. 
_V
3534elpw 3963 . . . . . . . . . . 11  |-  ( ( F `  v )  e.  ~P x  <->  ( F `  v )  C_  x
)
3633, 35sylibr 214 . . . . . . . . . 10  |-  ( v  e.  om  ->  ( F `  v )  e.  ~P x )
37 eleq1 2476 . . . . . . . . . 10  |-  ( ( F `  v )  =  u  ->  (
( F `  v
)  e.  ~P x  <->  u  e.  ~P x ) )
3836, 37syl5ibcom 222 . . . . . . . . 9  |-  ( v  e.  om  ->  (
( F `  v
)  =  u  ->  u  e.  ~P x
) )
3938rexlimiv 2892 . . . . . . . 8  |-  ( E. v  e.  om  ( F `  v )  =  u  ->  u  e. 
~P x )
4031, 39syl6bi 230 . . . . . . 7  |-  ( F  Fn  om  ->  (
u  e.  ran  F  ->  u  e.  ~P x
) )
4140ssrdv 3450 . . . . . 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 5575 . . . 4  |-  ( F : om --> ~P x  <->  ( F  Fn  om  /\  ran  F  C_  ~P x
) )
4543, 44mpbir 211 . . 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 6149 . 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 214 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 186    \/ wo 368    /\ wa 369    = wceq 1407    e. wcel 1844    =/= wne 2600   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3061    i^i cin 3415    C_ wss 3416    C. wpss 3417   (/)c0 3740   ~Pcpw 3957   U.cuni 4193    |-> cmpt 4455   ran crn 4826    |` cres 4827   Ord word 5411    Fn wfn 5566   -->wf 5567   -1-1->wf1 5568   ` cfv 5571   omcom 6685   reccrdg 7114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-reg 8054
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-om 6686  df-wrecs 7015  df-recs 7077  df-rdg 7115
This theorem is referenced by:  inf3lem7  8086  dominf  8859  dominfac  8982
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